Research ArticleControl Scheme for a Fractional-Order ChaoticGenesio-Tesi Model

ChangjinXu 1PeiluanLi 2MaoxinLiao3ZixinLiu 4QimeiXiao 5andShuaiYuan6

1Guizhou Key Laboratory of Economics System Simulation Guizhou University of Finance and EconomicsGuiyang 550004 China2School of Mathematics and Statistics Henan University of Science and Technology Luoyang 471023 China3School of Mathematics and Physics University of South China Hengyang 421001 China4School of Mathematics and Statistics Guizhou University of Finance and Economics Guiyang 550004 China5Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in EngineeringChangsha University of Science and Technology Changsha 410114 China6School of Mathematics and Statistics Central South University Changsha 410083 China

Correspondence should be addressed to Changjin Xu xcj403126com

Received 26 May 2019 Revised 24 July 2019 Accepted 3 September 2019 Published 29 September 2019

Academic Editor Chittaranjan Hens

Copyright copy 2019 Changjin Xu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper based on the earlier research a new fractional-order chaotic Genesio-Tesi model is established e chaoticphenomenon of the fractional-order chaotic Genesio-Tesi model is controlled by designing two suitable time-delayed feedbackcontrollers With the aid of Laplace transform we obtain the characteristic equation of the controlled chaotic Genesio-Tesimodel en by regarding the time delay as the bifurcation parameter and analyzing the characteristic equation some newsucient criteria to guarantee the stability and the existence of Hopf bifurcation for the controlled fractional-order chaoticGenesio-Tesi model are derived e research shows that when time delay remains in some interval the equilibrium point ofthe controlled chaotic Genesio-Tesi model is stable and a Hopf bifurcation will happen when the time delay crosses a criticalvalue e eect of the time delay on the stability and the existence of Hopf bifurcation for the controlled fractional-orderchaotic Genesio-Tesi model is shown At last computer simulations check the rationalization of the obtained theoreticalprediction e derived key results in this paper play an important role in controlling the chaotic behavior of many otherdierential chaotic systems

1 Introduction

As is known to us chaos control issue has been widelystudied in the last decades because of its potential prac-tical value in various areas How to control the chaoticphenomenon to serve human beings has become a hotissue in todayrsquos world Chaos control has attracted muchattention of researchers from various elds In recentyears many chaos control techniques (for exampledelayed feedback approach [1] OttndashGrebogindashYork (OGY)technique [2] variable structure control [3] observer-based control [4] backstepping design technique [5] andactive control [6]) have been proposed Many excellentfruits have been reported For example Kocamaz et al [7]

investigated the chaos control by applying the slidingmode control technique Chen [8] proposed an adaptivefeedback control technique for chaos and hyperchaoscontrol Din [9] studied the chaos control of a discrete-time prey-predator system by using three dierent typesof feedback control strategies Singh and Gakkhar [10]controlled the chaos of a food chain model with a time-delayed feedback controller Yan et al [11] discussed thechaos control of continuous unied chaotic systems ap-plying discrete rippling sliding mode control For moreknowledge on chaos control readers can refer to[12ndash19 34ndash44]

In 1992 Genesio and Tesi [20] put up the chaoticGenesio-Tesi system

HindawiComplexityVolume 2019 Article ID 4678394 15 pageshttpsdoiorg10115520194678394

dw1

dt w2

dw2

dt w3

dw3

dt k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where wi(i 1 2 3) denote different voltage of electroniccomponents and k1 k2 and k3 are all negative coefficientsGenesio and Tesi [20] investigated the chaotic behavior ofmodel (1) by applying harmonic balance approaches In2012 Guan et al [21] discussed the chaos control of model(1) by designing distributed delay feedback controller Byanalyzing the characteristic equation of the controlledGenesio-Tesi model the sufficient conditions to ensurethe stability of the equilibrium point and the existence ofHopf bifurcation for the controlled Genesio-Tesi modelare established In addition the stability and the directionof bifurcating periodic solution are determined by thecentre manifold theorem and normal form theory In2009 Sun [22] proposed a tracking control to realizechaos synchronization for the Genesio-Tesi chaotic sys-tem (1) based on the time-domain approach In 2007 Park[23] designed a novel feedback controller to realize ex-ponential synchronization of the Genesio-Tesi chaoticsystem (1) In 2009 Park [24] further considered thefunctional projective synchronization problem for theGenesio-Tesi chaotic system (1) and Zhou and Chen [25]dealt with the Hopf bifurcation and Sirsquolnikov chaos ofGenesio model (1)

It is worth mentioning that all the above papers focusonly on the integer-order differential models In recentyears many researchers argue that fractional-order differ-ential equations play a key role in describing the realphenomena and are found to have potential applications invarious areas such as physics economics physics heattransfer and chemical engineering [26ndash28] +us it is im-portant to deal with the fractional-order differential modelsBased on discussion above we revise model (1) in the fol-lowing fractional-order form

dσw1

dtσ w2

dσw2

dtσ w3

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where wi(i 1 2 3) denote different voltage of electroniccomponents k1 k2 and k3 are all negative coefficients andσ isin (0 1) denotes the fractional order Model (2) is thecommensurate fractional-order system Let σ 087 andk1 minus 6 k2 minus 292 and k3 minus 12 then the chaotic phe-nomenon will occur in model (2) +e fact is shown in thefollowing (see Figures 1 and 2)

+is paper mainly focuses on two aspects (a) designingtwo appropriate controllers to control the chaotic phe-nomenon of model (2) and (b) revealing the impact of timedelay on the stability and bifurcation behavior of the con-trolled fractional-order Genesio-Tesi chaotic model +esuperiority of this paper can be summarized as follows

(a) A new fractional-order Genesio-Tesi chaotic modelis proposed

(b) Two controllers are designed to control the chaoticphenomenon of the fractional-order Genesio-Tesichaotic model

(c) +e advantages and disadvantages of two controllersare compared

+e outline of this paper is organized as follows InSection 2 some elementary knowledge on fractional-orderdifferential is prepared In Section 3 two different controltechniques are designed to control the chaotic phenomenonof the fractional-order chaotic Genesio-Tesi model InSection 4 two examples are given to illustrate the theoreticalanalysis In Section 5 we end this manuscript withconclusion

2 Elementary Knowledge

In this section we list several basic results about fractional-order differential equations

Definition 1 (see [29]) +e Caputo fractional-order de-rivative is defined as follows

Dσq(l)

1Γ(n minus σ)

1113946l

l0

q(m)(s)

(l minus s)σminus m+1 ds (3)

where q(l) isin ([l0infin) R) lge l0 and m is a positive integersuch that σ isin [m minus 1 m) If σ isin (0 1) then

Dσq(l)

1Γ(1 minus σ)

1113946l

l0

qprime(s)

(l minus s)σds (4)

Definition 2 (see [30]) +e fractional-order system is givenas follows

dσw

dtσ g(w) w(0) w0 (5)

where σ isin (0 1] and w isin Rn wlowast is said to the equilibriumpoint of system (5) if g(wlowast) 0

Lemma 1 (see [31]) Denote λi(i 1 2 n) the root of thecharacteristic equation of the autonomous system

Dσw(t) Aw w(0) w0 (6)

where σ isin (0 1) w isin Rn andA isin Rntimesn 3en system (6) isasymptotically stable and also stable ⟺ |arg(λi)|

ge ((σπ)2)(i 1 2 n) and those critical eigenvalues thatsatisfy |arg(λi)| ((σπ)2)(i 1 2 n) have geometricmultiplicity one

Lemma 2 (see [32]) For the system

2 Complexity

0 50 100 150 200minus3minus2minus1

0123456

t

w 1 (t

)

(a)

0 50 100 150 200t

minus6

minus4

minus2

0

2

4

6

8

w 2 (t

)

(b)

0 50 100 150 200t

minus15

minus10

minus5

0

5

10

15

w 3 (t

)

(c)

minus3 minus2 minus1 0 1 2 3 4 5 6minus6

minus4

minus2

0

2

4

6

8

w 2 (t

)

w1 (t)

(d)

minus3 minus2 minus1 0 1 2 3 4 5 6minus15

minus10

minus5

0

5

10

15

w1 (t)

w 3 (t

)

(e)

w2 (t)

minus15

minus10

minus5

0

5

10

15

w 3 (t

)

minus6 minus4 minus2 0 2 4 6 8

(f )

0 50 100150

200

minus50

510

minus10

minus5

0

5

10

t

w 2 (t

)

w1 (t)

(g)

w 3 (t

)

minus4 minus2 0 2 4 6

minus10minus5

05

10minus15minus10

minus505

1015

w2 (t)

w1 (t)

(h)

Figure 1 Dynamics of system (2) when σ 087 and k1 minus 6 k2 minus 292 and k3 minus 12

Complexity 3

Dσw(t) Bw(t) + Cw(t minus ρ) (7)

where σ isin (0 1] w isin Rn andB C isin Rntimesn the characteristicequation takes the following form

det sσI minus B minus Ce

minus sρ11138681113868111386811138681113868111386811138681113868 0 (8)

+e zero solution of system (7) is asymptotically stable ifall the roots of equation (8) are negative real roots

Remark 1 In Lemma 1 and Lemma 2 we say that system (6)(or system (7)) is asymptotically stable (or stable) implyingthat the fixed point E of system (6) (or system (7)) is as-ymptotically stable (or stable)

3 Chaos Control via Time-DelayedFeedback Controllers

It is easy to see that system (2) has two unique equilibriumpoints

E1 (0 0 0)

E2 minus k1 0 0( 1113857(9)

A long time ago there were many publications thathandle the chaos control issue of integer-order differentialmodels by applying time-delayed feedback controllers Butthe studies on the chaos control of fractional-order chaoticmodels via time-delayed feedback controllers are very rareTo make up the deficiency in this paper we will design twotime-delayed feedback controllers to eliminate the chaotic

behavior of system (2) In this paper we are concerned onlywith the equilibrium point E1 (0 0 0) +e other equi-librium point E2 (minus k1 0 0) is easily analyzed in a similarway

31 Adding Time-Delayed Feedback Controller to the FirstEquation ofModel (2) In this section we add a time-delayedfeedback controller α1[w1(t) minus w1(t minus 9)] to the firstequation of model (2) and then system (2) takes the fol-lowing form

dσw1

dtσ w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

dσw2

dtσ w3

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

where α1 is the gain coefficient and 9 is the time delay +elinear system of system (10) near the equilibrium point E1

(0 0 0) readsdσw1

dtσ α1w1 + w2 minus α1w1(t minus 9)

dσw2

dtσ w3

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

+e corresponding characteristic equation of (11) takesthe following form

det

sσ minus α1 + α1eminus s9 minus 1 0

0 sσ minus 1

minus k1 minus k2 sσ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

Hence

A1(s) + A2(s)eminus s9

0 (13)

where

A1(s) s3σ

minus α1 + k3( 1113857s2σ

+ α1k3 minus k2( 1113857sσ

+ k2α1 minus k1

A2(s) α1s2σ

minus α1k3sσ

minus k2α1(14)

Suppose that s iχ χ(cos(π2) + i sin(π2)) is the rootof (13) then we get

A2R(χ)cos χ9 + A2I(ϕ)sin χ9 minus A1R(χ)

A2I(χ)cos χ9 minus A2R(ϕ)sin χ9 minus A1I(χ)1113896 (15)

where

02 03 04 05 06 07 08 090

02

04

06

08

1

12

14

16

18

Fractional order

Max

imum

Lya

puno

v ex

pone

nt

Figure 2 Maximum Lyapunov exponent of system (2) with respectto the fractional order σ

4 Complexity

A1R(χ) χ3σ cos3σπ2

minus α1 + k3( 1113857χ2σ cos σπ

+ α1k3 minus k2( 1113857χσ cosσπ2

+ k2α1 minus k1

A1I(χ) χ3σ sin3σπ2

minus α1 + k3( 1113857χ2σ sin σπ

+ α1k3 minus k2( 1113857χσ sinσπ2

A2R(χ) α1χ2σ cos σπ minus α1k3χσ cosσπ2

minus k2α1

A2I(χ) α1χ2σ sin σπ minus α1k3χσ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In view of (15) we have

cos χσ minusA1R(χ)A2R(χ) + A1I(χ)A2I(χ)

A22R(χ) + A2

2I(χ)

sin χσ minusA1R(χ)A2I(χ) minus A1I(χ)A2R(χ)

A22R(χ) + A2

2I(χ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

Let

p1 cos3σπ2

p2 minus α1 + k3( 1113857cos σπ

p3 α1k3 minus k2( 1113857cosσπ2

p4 k2α1 minus k1

p5 sin3σπ2

p6 minus α1 + k3( 1113857sin σπ

p7 α1k3 minus k2( 1113857sinσπ2

p8 α1 cos σπ

p9 minus α1k3 cosσπ2

p10 minus k2α1

p11 α1 sin σπ

p12 minus α1k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

HenceA1R(χ) p1χ3σ + p2χ2σ + p3χσ + p4

A1I(χ) p5χ3σ + p6χ2σ + p7χσ

A2R(χ) p8χ2σ + p9χσ + p10

A2I(χ) p11χ2σ + p12χσ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

By (19) we have

A1R(χ)A2R(χ) + A1I(χ)A2I(χ)1113858 11138592

+ A1R(χ)A2I(χ) minus A1I(χ)A2R(χ)1113858 11138592

A22R(χ) + A

22I(χ)1113960 1113961

2

(20)

In addition

A1R(χ)A2R(χ) + A1I(χ)A2I(χ)1113858 11138592

q1χ5σ

+ q2χ4σ

+ q3χ3σ

+ q4χ2σ

+ q5χσ

+ q61113872 11138732

A1R(χ)A2I(χ) minus A1I(χ)A2R(χ)1113858 11138592

q7χ5σ

+ b8χ4σ

+ q9χ3ϑ

+ q10χ2σ

+ q11χσ

1113872 11138732

A22R(χ) + A

22I(χ)1113960 1113961

2 q12χ

4q+ q13χ

3ϑ+ q14χ

2σ+ q15χ

σ1113872 1113873

2

(21)

whereq1 p1p8 + p5p11

q2 p2p8 + p1p9 + p6p11 + p5p12

q3 p3p8 + p2p9 + p1p10 + p7p11 + p6p12

q4 p4p8 + p3p9 + p2p10 + p7p12

q5 p4p9 + p3p10

q6 p4p10

q7 p1p11 minus p5p8

q8 p2p11 + p1p12 minus p5p9 minus p6p8

q9 p3p11 + p2p12 minus p5p10 minus p6p9 minus p7p8

q10 p4p11 + p3p12 minus p6p10 minus p7p9

q11 p4p12 minus p7p10

q12 p28 + p2

11

q13 2 p8p9 + p11p12( 1113857

q14 2p8p10 + p212

q15 2p9p10

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

By (20) we have

ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(23)

Complexity 5

whereξ1 q21 + q27

ξ2 2q1q2 + 2q7q8

ξ3 q22 + b28 minus q212 + 2q1q3 + 2q7q9

ξ4 2 q2q3 + q1q2 + q7q10 + q8q9 minus q12q13( 1113857

ξ5 q23 + 2q1q5 + 2q2q4 + q29 + 2q7q11 + 2q8q10 minus q213 minus 2q12q14

ξ6 2q1q6 + 2q2q5 + 2q3q4 + 2q9q10 minus 2q12q15 minus 2q13q14

ξ7 q24 + 2q3q5 + q210 + 2q8q11 + 2q9q11 minus q214 minus 2q13q15

ξ8 2q3q6 + 2q4q5 + 2q10q11 minus 2q14q15

ξ9 q25 + 2q4q6 + q211 minus q215

ξ10 2q5q6

ξ11 q26

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

Let

W(χ) ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(25)

V(μ) ξ1μ10

+ ξ2μ9

+ ξ3μ8

+ ξ4μ7

+ ξ5μ6

+ ξ6μ5

+ ξ7μ4

+ ξ8μ3

+ ξ9μ2

+ ξ10μ + ξ11(26)

Lemma 3

(i) Assume that ξj gt 0(j 1 2 11) then the root that haszero real parts does not exist in (13)(ii) Assume that ξ11 lt 0 and existμ0 gt 0 which satisfies V(μ0)lt 0then (13) possesses at least two pairs of purely imaginary roots

Proof of Lemma 3(i) According to (25) we havedW(χ)

dχ 10σξ1χ

10σminus 1+ 9σξ2χ

9σminus 1+ 8σξ3χ

8σminus 1+ 7σξ4χ

7σminus 1

+ 6σξ5χ6σminus 1

+ 5σξ6χ5σminus 1

+ 4σξ7χ4σminus 1

+ 3σξ8χ3σminus 1

+ 2σξ9χ2σminus 1

+ σξ10χσminus 1

(27)

In view of ξi gt 0(i 1 2 10) we have ((dW(χ))dχ)gt 0 forall χ gt 0 In view of W(0) ξ11 gt 0 we know that (26)does not possess positive real roots Furthermore s 0 is notthe root of (13) +is proves Lemma 3(i)

Proof of Lemma 3(ii) According to V(0) ξ11 gt0 V(ε0)lt 0(ε0 gt 0) and limτ⟶+infin(V(τ))dτ +infinexist ε01 isin (0 ε0) and ε02 isin (ε0 +infin) such that V(ε01)

V(ε02) 0 Hence (26) possesses at least two positive realroots and (13) possesses at least two pairs of purely imag-inary roots +is proves Lemma 3 (ii)

Suppose that (23) possesses ten positive real rootsζj(j 1 2 10) In view of (17) we get

9hm

1χm

arccos minusA1R χm( 1113857A2R χm( 1113857 + A1I χm( 1113857A2I χm( 1113857

A22R χm( 1113857 + A2

2I χm( 11138571113888 1113889 + 2hπ1113890 1113891

(28)

where h 0 1 2 andm 1 2 10 Hence plusmn iχm is apair of purely imaginary roots of (13) when 9 9h

m Let

90 minm1211

90m1113966 1113967

χ0 χ|990

(29)

Next the hypothesis H1 is given K1S1 + K2S2 gt 0

where

K1 3σχ3σminus 10 cos

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 cos

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 cos

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891cos χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891sin χ090

K2 3σχ3σminus 10 sin

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 sin

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 sin

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891sin χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891cos χ090

S1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

S2 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 cos χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 sin χ090

(30)

Lemma 4 Suppose that s(9) c(9) + iϕ(9) is the root of(13) around 9 90 such that c(90) 0 ϕ(90) ϕ0 thenRe[dsd9]|990 ϕϕ0 gt 0

Proof of Lemma 4 In view of (13) we get

dA1(s)

d9+

dA2(s)

d9e

minus s9minus e

minus s9 ds

d99 + s1113888 1113889A2(s) 0 (31)

6 Complexity

SincedA1(s)

d9 3σs

3σminus 1ds

d9minus 2σ α1 + k3( 1113857s

2σminus 1ds

d9+ σ α1k3 minus k2( 1113857s

σminus 1ds

d9

3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

1113960 1113961ds

d9

dA2(s)

d9 2α1σs

2σminus 1minus α1k3σs

σminus 11113872 1113873

ds

d9

(32)

we obtainds

d91113890 1113891

minus 1

P1(9)

P2(9)minus

9

s (33)

where

P1(s) 3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

+ 2α1σs2σminus 1

minus α1k3σsσminus 1

1113872 1113873eminus s9

P2(s) A2(s)seminus s9

(34)

+us

Reds

d91113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

ReP1(9)

P2(9)1113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

K1S1 + K2S2

S21 + S2

2

(35)

It follows from H1 that

Reds

d91113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ψψ0

gt 0 (36)

+is proves Lemma 4

Next the hypothesis H2 is given

k3 lt 0

k2k3 gt minus 2k1

k1 lt 0

(37)

Lemma 5 If 9 0 and H2 holds true then system (13) islocally asymptotically stable

Proof of Lemma 5 Under the condition 9 0 (13) becomes

λ3 minus k3λ2

minus k2λ minus k1 0 (38)

In view of H2 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (38) +us Lemma 5 holds true implying theproof

+e following results are established by above discussion

Theorem 1 (under the condition of Lemma 3(ii)) Supposethat H1 and H2 hold true then (i) the equilibrium point of

system (10) at the origin is locally asymptotically stable if0le 9lt 90 and (ii) a Hopf bifurcation will happen near theequilibrium point E1 if 9 90

32 Adding Time-Delayed Feedback Controller to the Sec-ond Equation of Model (2) In this section we add a time-delayed feedback controller α2[w2(t) minus w2(t minus ρ)] tothe second equation of model (2) and then system (2)becomes

dσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)1113858 1113859

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(39)

where α2 is the gain coefficient and ρ is the time delay +elinear equation of equation (39) around E1 (0 0 0) is

dσw1

dtσ w2

dσw2

dtσ α2w2 + w3 minus α2w2(t minus ρ)

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

+e characteristic equation of (40) takes the followingform

det

sσ minus 1 0

0 sσ minus α2 + α2eminus sρ minus 1

minus k11 minus k2 sρ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (41)

Hence

B1(s) + B2(s)eminus sρ

0 (42)

where

B1(s) s3σ

minus α2 + k3 + 1( 1113857s2σ

+ α2k3 + k3 minus k2( 1113857sσ

minus k1

B2(s) α2s2σ

minus α2k3sσ

(43)

Assume that s iϑ ϑ(cos(π2) + i sin(π2)) is the rootof (42) then we have

B2R(ϑ)cos ϑρ + B2I(ϑ)sin ζρ minus B1R(ϑ)

B2I(ϑ)cos ϑρ minus B2R(ϕ)sin ϑρ minus B1I(ϑ)1113896 (44)

where

Complexity 7

B1R(ϑ) ϑ3σ cos3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ cos σπ

+ α2k3 + k3 minus k2( 1113857ϑσ cosσπ2

minus k1

B1I(ϑ) ϑ3σ sin3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ sin σπ

+ α2k3 + k3 minus k2( 1113857ϑσ sinσπ2

B2R(ϑ) α2ϑ2ϑ cos ϑπ minus α2k3ϑ

σ cosσπ2

B2I(ϑ) α2ϑ2ϑ sin ϑπ minus α2k3ϑ

σ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

According to (44) we get

cos ϑρ minusB1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)

B22R(ϑ) + B2

2I(ϑ)

sin ϑρ minusB1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)

B22R(ϑ) + B2

2I(ϑ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

Let

s1 cos3σπ2

s2 minus α2 + k3 + 1( 1113857cos σπ

s3 α2k3 + k3 minus k2( 1113857cosσπ2

s4 minus k1

s5 sin3σπ2

s6 minus α2 + k3 + 1( 1113857sin σπ

s7 α2k3 + k3 minus k2( 1113857sinσπ2

s8 α2 cos ϑπ

s9 minus α2k3 cosσπ2

s10 α2 sin ϑπ

s11 minus α2k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

+en

B1R(ϑ) s1ϑ3σ + s2ϑ

2σ + s3ϑσ + s4

B1I(ϑ) s5ϑ3σ + s6ϑ

2σ + s7ϑσ

B2R(ϑ) s8ϑ2σ + s9ϑ

σ

B2I(ϑ) s10ϑ2σ + s11ϑ

σ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(48)

By (46) we have

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

+ B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

B22R(ϑ) + B

22I(ϑ)1113960 1113961

2

(49)

Furthermore

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

t1ϑ5σ

+ t2ϑ4σ

+ t3ϑ3σ

+ t4ϑ2σ

+ t5ϑσ

1113872 11138732

B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

t5ϑ5σ

+ t6ϑ4σ

+ t7ϑ3ϑ

+ t8ϑ2σ

+ t9ϑσ

1113872 11138732

B22R(ϑ) + B

22I(χ)1113960 1113961

2 t10ϑ

4σ+ t11ϑ

3σ+ t12ϑ

2σ+ t13ϑ

σ1113872 1113873

2

(50)

wheret1 s1s8 + s5s10

t2 s2s8 + s1s9 + s6s10 + s5s11

t3 s3s8 + s2s9 + s7s10 + s6s11

t4 s4s8 + s3s9 + s7s11

t5 s4s9

t6 s1s10 minus s5s8

t7 s2s10 + s1s11 minus s5s9 minus s6s8

t8 s3s10 + s2s11 minus s6s9 minus s7s8

t9 s4s10 + s3s11 minus s7s9

t10 s4s11

t11 s28 + s210

t12 2 s8s9 + s10s11( 1113857

t13 s211

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(51)

In view of (49) we have

κ1ϑ10σ

+ κ2ϑ9σ

+ κ3ϑ8σ

+ κ4ϑ7σ

+ κ5ϑ6σ

+ κ6ϑ5σ

+ κ7ϑ4σ

+ κ8ϑ3σ

+ κ9ϑ2σ

0

(52)

where

8 Complexity

κ1 t21 + t26

κ2 2t1t2 + 2t6t7

κ3 t22 + t27 minus t211 + 2t1t3 + 2t6t8

κ4 2 t2t3 + t1t2 + t6t9 + t7t8 minus t11t12( 1113857

κ5 t23 + 2t2t4 + t28 + 2t6t10 + 2t7t9 minus t212 minus 2t11t13

κ6 2t3t4 + 2t8t9

κ7 t24 + t29 + 2t7t10 + 2t8t10 minus t213

κ8 2t9t10 + 2t4t5

κ9 t25 + t210

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(53)

By (52) we have

κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9 0

(54)

Let

K(ϑ) κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9(55)

M(ζ) κ1ζ8

+ κ2ζ7

+ κ3ζ6

+ κ4ζ5

+ κ5ζ4

+ κ6ζ3

+ κ7ζ2

+ κ8ζ1

+ κ9(56)

Lemma 6

(i) Suppose that κj gt 0(j 1 2 9) then the root that haszero real parts does not exist in (42)(ii) Suppose that κ9 lt 0 and existη0 gt 0 which satisfies M(η0)lt 0then (42) possesses at least two pairs of purely imaginary roots

Proof of Lemma 6(i) In view of (55) we havedK(ϑ)

dχ 8κ1σϑ

8σminus 1+ 7κ2σϑ

7σminus 1+ 6κ3σϑ

6σminus 1+ 5κ4σϑ

5σminus 1

+ 4κ5σϑ4σminus 1

+ 3κ6σϑ3σminus 1

+ 2κ7σϑ2σminus 1

+ κ8σϑσminus 1

(57)

In view of κi gt 0(i 1 2 9) we have((dK(ϑ))dϑ) gt 0 forall ϑgt 0 In view of K(0) κ9 gt 0 we knowthat (56) does not possess positive real roots Furthermores 0 is not the root of (42) +is proves Lemma 6(i)

Proof of Lemma 6(ii) In view of M(0) κ9 gt 0 M(η0)lt0(η0 gt 0) and limε⟶+infin((M(ε))dε) +infin exist η01 isin (0 η0)and η02 isin (η0 +infin) such that M(η01) M(η02) 0 Hence(55) has at least two positive real roots and (42) has at leasttwo pairs of purely imaginary roots +is proves Lemma6(ii)

Assume that (54) has eight positive real rootsϑj(j 1 2 8) By (46) we have

ριk 1ϑk

arccos minusB1R ϑm( 1113857B2R ϑk( 1113857 + B1I ϑk( 1113857B2I ϑk( 1113857

B22R ϑk( 1113857 + B2

2I ϑk( 11138571113888 1113889 + 2ιπ1113890 1113891

(58)

where ι 0 1 2 and k 1 2 8 Hence plusmn iϑk is apair of purely imaginary roots of (42) when ρ ριk Let

ρ0 mink128

ρ0k1113966 1113967

ϑ0 ϑ|ρρ0(59)

Next we give the hypothesis H3 as followsF1G1 + F2G2 gt 0 where

F1 3σϑ3σminus 10 cos

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 cos

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 cos

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

F2 3σϑ3σminus 10 sin

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 sin

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 sin

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

G1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

G2 α2ϑ2σ0 cos σπ minus α2k3χ

σ0 cos

σπ2

minus α2k31113876 1113877χ0 cos ϑ0ρ0

+ α2ϑ2σ0 sin σπ minus α2k3ϑ

σ0 sin

σπ2

minus α2k31113876 1113877ϑ0 sin ϑ0ρ0

(60)

Lemma 7 Assume that s(ρ) ζ(ρ) + iϖ(ρ) is the root of(42) around ρ ρ0 such that ζ(ρ0) 0 ϖ(ρ0) ϖ0 thenRe[dsdρ]|ρρ0 ϖϖ0 gt 0

Proof of Lemma 7 By (42) we have

dB1(s)

dρ+

dB2(s)

dρe

minus sρminus e

minus sρ ds

dρρ + s1113888 1113889B2(s) 0 (61)

Complexity 9

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

0 50 100 150 200minus3minus2minus1

0123456

t

w 1 (t

)

(a)

0 50 100 150 200t

minus6

minus4

minus2

0

2

4

6

8

w 2 (t

)

(b)

0 50 100 150 200t

minus15

minus10

minus5

0

5

10

15

w 3 (t

)

(c)

minus3 minus2 minus1 0 1 2 3 4 5 6minus6

minus4

minus2

0

2

4

6

8

w 2 (t

)

w1 (t)

(d)

minus3 minus2 minus1 0 1 2 3 4 5 6minus15

minus10

minus5

0

5

10

15

w1 (t)

w 3 (t

)

(e)

w2 (t)

minus15

minus10

minus5

0

5

10

15

w 3 (t

)

minus6 minus4 minus2 0 2 4 6 8

(f )

0 50 100150

200

minus50

510

minus10

minus5

0

5

10

t

w 2 (t

)

w1 (t)

(g)

w 3 (t

)

minus4 minus2 0 2 4 6

minus10minus5

05

10minus15minus10

minus505

1015

w2 (t)

w1 (t)

(h)

Figure 1 Dynamics of system (2) when σ 087 and k1 minus 6 k2 minus 292 and k3 minus 12

Complexity 3

Dσw(t) Bw(t) + Cw(t minus ρ) (7)

where σ isin (0 1] w isin Rn andB C isin Rntimesn the characteristicequation takes the following form

det sσI minus B minus Ce

minus sρ11138681113868111386811138681113868111386811138681113868 0 (8)

+e zero solution of system (7) is asymptotically stable ifall the roots of equation (8) are negative real roots

Remark 1 In Lemma 1 and Lemma 2 we say that system (6)(or system (7)) is asymptotically stable (or stable) implyingthat the fixed point E of system (6) (or system (7)) is as-ymptotically stable (or stable)

3 Chaos Control via Time-DelayedFeedback Controllers

It is easy to see that system (2) has two unique equilibriumpoints

E1 (0 0 0)

E2 minus k1 0 0( 1113857(9)

A long time ago there were many publications thathandle the chaos control issue of integer-order differentialmodels by applying time-delayed feedback controllers Butthe studies on the chaos control of fractional-order chaoticmodels via time-delayed feedback controllers are very rareTo make up the deficiency in this paper we will design twotime-delayed feedback controllers to eliminate the chaotic

behavior of system (2) In this paper we are concerned onlywith the equilibrium point E1 (0 0 0) +e other equi-librium point E2 (minus k1 0 0) is easily analyzed in a similarway

31 Adding Time-Delayed Feedback Controller to the FirstEquation ofModel (2) In this section we add a time-delayedfeedback controller α1[w1(t) minus w1(t minus 9)] to the firstequation of model (2) and then system (2) takes the fol-lowing form

dσw1

dtσ w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

dσw2

dtσ w3

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

where α1 is the gain coefficient and 9 is the time delay +elinear system of system (10) near the equilibrium point E1

(0 0 0) readsdσw1

dtσ α1w1 + w2 minus α1w1(t minus 9)

dσw2

dtσ w3

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

+e corresponding characteristic equation of (11) takesthe following form

det

sσ minus α1 + α1eminus s9 minus 1 0

0 sσ minus 1

minus k1 minus k2 sσ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

Hence

A1(s) + A2(s)eminus s9

0 (13)

where

A1(s) s3σ

minus α1 + k3( 1113857s2σ

+ α1k3 minus k2( 1113857sσ

+ k2α1 minus k1

A2(s) α1s2σ

minus α1k3sσ

minus k2α1(14)

Suppose that s iχ χ(cos(π2) + i sin(π2)) is the rootof (13) then we get

A2R(χ)cos χ9 + A2I(ϕ)sin χ9 minus A1R(χ)

A2I(χ)cos χ9 minus A2R(ϕ)sin χ9 minus A1I(χ)1113896 (15)

where

02 03 04 05 06 07 08 090

02

04

06

08

1

12

14

16

18

Fractional order

Max

imum

Lya

puno

v ex

pone

nt

Figure 2 Maximum Lyapunov exponent of system (2) with respectto the fractional order σ

4 Complexity

A1R(χ) χ3σ cos3σπ2

minus α1 + k3( 1113857χ2σ cos σπ

+ α1k3 minus k2( 1113857χσ cosσπ2

+ k2α1 minus k1

A1I(χ) χ3σ sin3σπ2

minus α1 + k3( 1113857χ2σ sin σπ

+ α1k3 minus k2( 1113857χσ sinσπ2

A2R(χ) α1χ2σ cos σπ minus α1k3χσ cosσπ2

minus k2α1

A2I(χ) α1χ2σ sin σπ minus α1k3χσ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In view of (15) we have

cos χσ minusA1R(χ)A2R(χ) + A1I(χ)A2I(χ)

A22R(χ) + A2

2I(χ)

sin χσ minusA1R(χ)A2I(χ) minus A1I(χ)A2R(χ)

A22R(χ) + A2

2I(χ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

Let

p1 cos3σπ2

p2 minus α1 + k3( 1113857cos σπ

p3 α1k3 minus k2( 1113857cosσπ2

p4 k2α1 minus k1

p5 sin3σπ2

p6 minus α1 + k3( 1113857sin σπ

p7 α1k3 minus k2( 1113857sinσπ2

p8 α1 cos σπ

p9 minus α1k3 cosσπ2

p10 minus k2α1

p11 α1 sin σπ

p12 minus α1k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

HenceA1R(χ) p1χ3σ + p2χ2σ + p3χσ + p4

A1I(χ) p5χ3σ + p6χ2σ + p7χσ

A2R(χ) p8χ2σ + p9χσ + p10

A2I(χ) p11χ2σ + p12χσ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

By (19) we have

A1R(χ)A2R(χ) + A1I(χ)A2I(χ)1113858 11138592

+ A1R(χ)A2I(χ) minus A1I(χ)A2R(χ)1113858 11138592

A22R(χ) + A

22I(χ)1113960 1113961

2

(20)

In addition

A1R(χ)A2R(χ) + A1I(χ)A2I(χ)1113858 11138592

q1χ5σ

+ q2χ4σ

+ q3χ3σ

+ q4χ2σ

+ q5χσ

+ q61113872 11138732

A1R(χ)A2I(χ) minus A1I(χ)A2R(χ)1113858 11138592

q7χ5σ

+ b8χ4σ

+ q9χ3ϑ

+ q10χ2σ

+ q11χσ

1113872 11138732

A22R(χ) + A

22I(χ)1113960 1113961

2 q12χ

4q+ q13χ

3ϑ+ q14χ

2σ+ q15χ

σ1113872 1113873

2

(21)

whereq1 p1p8 + p5p11

q2 p2p8 + p1p9 + p6p11 + p5p12

q3 p3p8 + p2p9 + p1p10 + p7p11 + p6p12

q4 p4p8 + p3p9 + p2p10 + p7p12

q5 p4p9 + p3p10

q6 p4p10

q7 p1p11 minus p5p8

q8 p2p11 + p1p12 minus p5p9 minus p6p8

q9 p3p11 + p2p12 minus p5p10 minus p6p9 minus p7p8

q10 p4p11 + p3p12 minus p6p10 minus p7p9

q11 p4p12 minus p7p10

q12 p28 + p2

11

q13 2 p8p9 + p11p12( 1113857

q14 2p8p10 + p212

q15 2p9p10

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

By (20) we have

ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(23)

Complexity 5

whereξ1 q21 + q27

ξ2 2q1q2 + 2q7q8

ξ3 q22 + b28 minus q212 + 2q1q3 + 2q7q9

ξ4 2 q2q3 + q1q2 + q7q10 + q8q9 minus q12q13( 1113857

ξ5 q23 + 2q1q5 + 2q2q4 + q29 + 2q7q11 + 2q8q10 minus q213 minus 2q12q14

ξ6 2q1q6 + 2q2q5 + 2q3q4 + 2q9q10 minus 2q12q15 minus 2q13q14

ξ7 q24 + 2q3q5 + q210 + 2q8q11 + 2q9q11 minus q214 minus 2q13q15

ξ8 2q3q6 + 2q4q5 + 2q10q11 minus 2q14q15

ξ9 q25 + 2q4q6 + q211 minus q215

ξ10 2q5q6

ξ11 q26

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

Let

W(χ) ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(25)

V(μ) ξ1μ10

+ ξ2μ9

+ ξ3μ8

+ ξ4μ7

+ ξ5μ6

+ ξ6μ5

+ ξ7μ4

+ ξ8μ3

+ ξ9μ2

+ ξ10μ + ξ11(26)

Lemma 3

(i) Assume that ξj gt 0(j 1 2 11) then the root that haszero real parts does not exist in (13)(ii) Assume that ξ11 lt 0 and existμ0 gt 0 which satisfies V(μ0)lt 0then (13) possesses at least two pairs of purely imaginary roots

Proof of Lemma 3(i) According to (25) we havedW(χ)

dχ 10σξ1χ

10σminus 1+ 9σξ2χ

9σminus 1+ 8σξ3χ

8σminus 1+ 7σξ4χ

7σminus 1

+ 6σξ5χ6σminus 1

+ 5σξ6χ5σminus 1

+ 4σξ7χ4σminus 1

+ 3σξ8χ3σminus 1

+ 2σξ9χ2σminus 1

+ σξ10χσminus 1

(27)

In view of ξi gt 0(i 1 2 10) we have ((dW(χ))dχ)gt 0 forall χ gt 0 In view of W(0) ξ11 gt 0 we know that (26)does not possess positive real roots Furthermore s 0 is notthe root of (13) +is proves Lemma 3(i)

Proof of Lemma 3(ii) According to V(0) ξ11 gt0 V(ε0)lt 0(ε0 gt 0) and limτ⟶+infin(V(τ))dτ +infinexist ε01 isin (0 ε0) and ε02 isin (ε0 +infin) such that V(ε01)

V(ε02) 0 Hence (26) possesses at least two positive realroots and (13) possesses at least two pairs of purely imag-inary roots +is proves Lemma 3 (ii)

Suppose that (23) possesses ten positive real rootsζj(j 1 2 10) In view of (17) we get

9hm

1χm

arccos minusA1R χm( 1113857A2R χm( 1113857 + A1I χm( 1113857A2I χm( 1113857

A22R χm( 1113857 + A2

2I χm( 11138571113888 1113889 + 2hπ1113890 1113891

(28)

where h 0 1 2 andm 1 2 10 Hence plusmn iχm is apair of purely imaginary roots of (13) when 9 9h

m Let

90 minm1211

90m1113966 1113967

χ0 χ|990

(29)

Next the hypothesis H1 is given K1S1 + K2S2 gt 0

where

K1 3σχ3σminus 10 cos

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 cos

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 cos

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891cos χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891sin χ090

K2 3σχ3σminus 10 sin

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 sin

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 sin

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891sin χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891cos χ090

S1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

S2 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 cos χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 sin χ090

(30)

Lemma 4 Suppose that s(9) c(9) + iϕ(9) is the root of(13) around 9 90 such that c(90) 0 ϕ(90) ϕ0 thenRe[dsd9]|990 ϕϕ0 gt 0

Proof of Lemma 4 In view of (13) we get

dA1(s)

d9+

dA2(s)

d9e

minus s9minus e

minus s9 ds

d99 + s1113888 1113889A2(s) 0 (31)

6 Complexity

SincedA1(s)

d9 3σs

3σminus 1ds

d9minus 2σ α1 + k3( 1113857s

2σminus 1ds

d9+ σ α1k3 minus k2( 1113857s

σminus 1ds

d9

3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

1113960 1113961ds

d9

dA2(s)

d9 2α1σs

2σminus 1minus α1k3σs

σminus 11113872 1113873

ds

d9

(32)

we obtainds

d91113890 1113891

minus 1

P1(9)

P2(9)minus

9

s (33)

where

P1(s) 3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

+ 2α1σs2σminus 1

minus α1k3σsσminus 1

1113872 1113873eminus s9

P2(s) A2(s)seminus s9

(34)

+us

Reds

d91113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

ReP1(9)

P2(9)1113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

K1S1 + K2S2

S21 + S2

2

(35)

It follows from H1 that

Reds

d91113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ψψ0

gt 0 (36)

+is proves Lemma 4

Next the hypothesis H2 is given

k3 lt 0

k2k3 gt minus 2k1

k1 lt 0

(37)

Lemma 5 If 9 0 and H2 holds true then system (13) islocally asymptotically stable

Proof of Lemma 5 Under the condition 9 0 (13) becomes

λ3 minus k3λ2

minus k2λ minus k1 0 (38)

In view of H2 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (38) +us Lemma 5 holds true implying theproof

+e following results are established by above discussion

Theorem 1 (under the condition of Lemma 3(ii)) Supposethat H1 and H2 hold true then (i) the equilibrium point of

system (10) at the origin is locally asymptotically stable if0le 9lt 90 and (ii) a Hopf bifurcation will happen near theequilibrium point E1 if 9 90

32 Adding Time-Delayed Feedback Controller to the Sec-ond Equation of Model (2) In this section we add a time-delayed feedback controller α2[w2(t) minus w2(t minus ρ)] tothe second equation of model (2) and then system (2)becomes

dσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)1113858 1113859

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(39)

where α2 is the gain coefficient and ρ is the time delay +elinear equation of equation (39) around E1 (0 0 0) is

dσw1

dtσ w2

dσw2

dtσ α2w2 + w3 minus α2w2(t minus ρ)

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

+e characteristic equation of (40) takes the followingform

det

sσ minus 1 0

0 sσ minus α2 + α2eminus sρ minus 1

minus k11 minus k2 sρ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (41)

Hence

B1(s) + B2(s)eminus sρ

0 (42)

where

B1(s) s3σ

minus α2 + k3 + 1( 1113857s2σ

+ α2k3 + k3 minus k2( 1113857sσ

minus k1

B2(s) α2s2σ

minus α2k3sσ

(43)

Assume that s iϑ ϑ(cos(π2) + i sin(π2)) is the rootof (42) then we have

B2R(ϑ)cos ϑρ + B2I(ϑ)sin ζρ minus B1R(ϑ)

B2I(ϑ)cos ϑρ minus B2R(ϕ)sin ϑρ minus B1I(ϑ)1113896 (44)

where

Complexity 7

B1R(ϑ) ϑ3σ cos3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ cos σπ

+ α2k3 + k3 minus k2( 1113857ϑσ cosσπ2

minus k1

B1I(ϑ) ϑ3σ sin3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ sin σπ

+ α2k3 + k3 minus k2( 1113857ϑσ sinσπ2

B2R(ϑ) α2ϑ2ϑ cos ϑπ minus α2k3ϑ

σ cosσπ2

B2I(ϑ) α2ϑ2ϑ sin ϑπ minus α2k3ϑ

σ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

According to (44) we get

cos ϑρ minusB1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)

B22R(ϑ) + B2

2I(ϑ)

sin ϑρ minusB1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)

B22R(ϑ) + B2

2I(ϑ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

Let

s1 cos3σπ2

s2 minus α2 + k3 + 1( 1113857cos σπ

s3 α2k3 + k3 minus k2( 1113857cosσπ2

s4 minus k1

s5 sin3σπ2

s6 minus α2 + k3 + 1( 1113857sin σπ

s7 α2k3 + k3 minus k2( 1113857sinσπ2

s8 α2 cos ϑπ

s9 minus α2k3 cosσπ2

s10 α2 sin ϑπ

s11 minus α2k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

+en

B1R(ϑ) s1ϑ3σ + s2ϑ

2σ + s3ϑσ + s4

B1I(ϑ) s5ϑ3σ + s6ϑ

2σ + s7ϑσ

B2R(ϑ) s8ϑ2σ + s9ϑ

σ

B2I(ϑ) s10ϑ2σ + s11ϑ

σ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(48)

By (46) we have

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

+ B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

B22R(ϑ) + B

22I(ϑ)1113960 1113961

2

(49)

Furthermore

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

t1ϑ5σ

+ t2ϑ4σ

+ t3ϑ3σ

+ t4ϑ2σ

+ t5ϑσ

1113872 11138732

B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

t5ϑ5σ

+ t6ϑ4σ

+ t7ϑ3ϑ

+ t8ϑ2σ

+ t9ϑσ

1113872 11138732

B22R(ϑ) + B

22I(χ)1113960 1113961

2 t10ϑ

4σ+ t11ϑ

3σ+ t12ϑ

2σ+ t13ϑ

σ1113872 1113873

2

(50)

wheret1 s1s8 + s5s10

t2 s2s8 + s1s9 + s6s10 + s5s11

t3 s3s8 + s2s9 + s7s10 + s6s11

t4 s4s8 + s3s9 + s7s11

t5 s4s9

t6 s1s10 minus s5s8

t7 s2s10 + s1s11 minus s5s9 minus s6s8

t8 s3s10 + s2s11 minus s6s9 minus s7s8

t9 s4s10 + s3s11 minus s7s9

t10 s4s11

t11 s28 + s210

t12 2 s8s9 + s10s11( 1113857

t13 s211

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(51)

In view of (49) we have

κ1ϑ10σ

+ κ2ϑ9σ

+ κ3ϑ8σ

+ κ4ϑ7σ

+ κ5ϑ6σ

+ κ6ϑ5σ

+ κ7ϑ4σ

+ κ8ϑ3σ

+ κ9ϑ2σ

0

(52)

where

8 Complexity

κ1 t21 + t26

κ2 2t1t2 + 2t6t7

κ3 t22 + t27 minus t211 + 2t1t3 + 2t6t8

κ4 2 t2t3 + t1t2 + t6t9 + t7t8 minus t11t12( 1113857

κ5 t23 + 2t2t4 + t28 + 2t6t10 + 2t7t9 minus t212 minus 2t11t13

κ6 2t3t4 + 2t8t9

κ7 t24 + t29 + 2t7t10 + 2t8t10 minus t213

κ8 2t9t10 + 2t4t5

κ9 t25 + t210

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(53)

By (52) we have

κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9 0

(54)

Let

K(ϑ) κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9(55)

M(ζ) κ1ζ8

+ κ2ζ7

+ κ3ζ6

+ κ4ζ5

+ κ5ζ4

+ κ6ζ3

+ κ7ζ2

+ κ8ζ1

+ κ9(56)

Lemma 6

(i) Suppose that κj gt 0(j 1 2 9) then the root that haszero real parts does not exist in (42)(ii) Suppose that κ9 lt 0 and existη0 gt 0 which satisfies M(η0)lt 0then (42) possesses at least two pairs of purely imaginary roots

Proof of Lemma 6(i) In view of (55) we havedK(ϑ)

dχ 8κ1σϑ

8σminus 1+ 7κ2σϑ

7σminus 1+ 6κ3σϑ

6σminus 1+ 5κ4σϑ

5σminus 1

+ 4κ5σϑ4σminus 1

+ 3κ6σϑ3σminus 1

+ 2κ7σϑ2σminus 1

+ κ8σϑσminus 1

(57)

In view of κi gt 0(i 1 2 9) we have((dK(ϑ))dϑ) gt 0 forall ϑgt 0 In view of K(0) κ9 gt 0 we knowthat (56) does not possess positive real roots Furthermores 0 is not the root of (42) +is proves Lemma 6(i)

Proof of Lemma 6(ii) In view of M(0) κ9 gt 0 M(η0)lt0(η0 gt 0) and limε⟶+infin((M(ε))dε) +infin exist η01 isin (0 η0)and η02 isin (η0 +infin) such that M(η01) M(η02) 0 Hence(55) has at least two positive real roots and (42) has at leasttwo pairs of purely imaginary roots +is proves Lemma6(ii)

Assume that (54) has eight positive real rootsϑj(j 1 2 8) By (46) we have

ριk 1ϑk

arccos minusB1R ϑm( 1113857B2R ϑk( 1113857 + B1I ϑk( 1113857B2I ϑk( 1113857

B22R ϑk( 1113857 + B2

2I ϑk( 11138571113888 1113889 + 2ιπ1113890 1113891

(58)

where ι 0 1 2 and k 1 2 8 Hence plusmn iϑk is apair of purely imaginary roots of (42) when ρ ριk Let

ρ0 mink128

ρ0k1113966 1113967

ϑ0 ϑ|ρρ0(59)

Next we give the hypothesis H3 as followsF1G1 + F2G2 gt 0 where

F1 3σϑ3σminus 10 cos

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 cos

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 cos

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

F2 3σϑ3σminus 10 sin

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 sin

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 sin

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

G1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

G2 α2ϑ2σ0 cos σπ minus α2k3χ

σ0 cos

σπ2

minus α2k31113876 1113877χ0 cos ϑ0ρ0

+ α2ϑ2σ0 sin σπ minus α2k3ϑ

σ0 sin

σπ2

minus α2k31113876 1113877ϑ0 sin ϑ0ρ0

(60)

Lemma 7 Assume that s(ρ) ζ(ρ) + iϖ(ρ) is the root of(42) around ρ ρ0 such that ζ(ρ0) 0 ϖ(ρ0) ϖ0 thenRe[dsdρ]|ρρ0 ϖϖ0 gt 0

Proof of Lemma 7 By (42) we have

dB1(s)

dρ+

dB2(s)

dρe

minus sρminus e

minus sρ ds

dρρ + s1113888 1113889B2(s) 0 (61)

Complexity 9

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Dσw(t) Bw(t) + Cw(t minus ρ) (7)

where σ isin (0 1] w isin Rn andB C isin Rntimesn the characteristicequation takes the following form

det sσI minus B minus Ce

minus sρ11138681113868111386811138681113868111386811138681113868 0 (8)

+e zero solution of system (7) is asymptotically stable ifall the roots of equation (8) are negative real roots

Remark 1 In Lemma 1 and Lemma 2 we say that system (6)(or system (7)) is asymptotically stable (or stable) implyingthat the fixed point E of system (6) (or system (7)) is as-ymptotically stable (or stable)

3 Chaos Control via Time-DelayedFeedback Controllers

It is easy to see that system (2) has two unique equilibriumpoints

E1 (0 0 0)

E2 minus k1 0 0( 1113857(9)

A long time ago there were many publications thathandle the chaos control issue of integer-order differentialmodels by applying time-delayed feedback controllers Butthe studies on the chaos control of fractional-order chaoticmodels via time-delayed feedback controllers are very rareTo make up the deficiency in this paper we will design twotime-delayed feedback controllers to eliminate the chaotic

behavior of system (2) In this paper we are concerned onlywith the equilibrium point E1 (0 0 0) +e other equi-librium point E2 (minus k1 0 0) is easily analyzed in a similarway

31 Adding Time-Delayed Feedback Controller to the FirstEquation ofModel (2) In this section we add a time-delayedfeedback controller α1[w1(t) minus w1(t minus 9)] to the firstequation of model (2) and then system (2) takes the fol-lowing form

dσw1

dtσ w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

dσw2

dtσ w3

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

where α1 is the gain coefficient and 9 is the time delay +elinear system of system (10) near the equilibrium point E1

(0 0 0) readsdσw1

dtσ α1w1 + w2 minus α1w1(t minus 9)

dσw2

dtσ w3

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

+e corresponding characteristic equation of (11) takesthe following form

det

sσ minus α1 + α1eminus s9 minus 1 0

0 sσ minus 1

minus k1 minus k2 sσ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (12)

Hence

A1(s) + A2(s)eminus s9

0 (13)

where

A1(s) s3σ

minus α1 + k3( 1113857s2σ

+ α1k3 minus k2( 1113857sσ

+ k2α1 minus k1

A2(s) α1s2σ

minus α1k3sσ

minus k2α1(14)

Suppose that s iχ χ(cos(π2) + i sin(π2)) is the rootof (13) then we get

A2R(χ)cos χ9 + A2I(ϕ)sin χ9 minus A1R(χ)

A2I(χ)cos χ9 minus A2R(ϕ)sin χ9 minus A1I(χ)1113896 (15)

where

02 03 04 05 06 07 08 090

02

04

06

08

1

12

14

16

18

Fractional order

Max

imum

Lya

puno

v ex

pone

nt

Figure 2 Maximum Lyapunov exponent of system (2) with respectto the fractional order σ

4 Complexity

A1R(χ) χ3σ cos3σπ2

minus α1 + k3( 1113857χ2σ cos σπ

+ α1k3 minus k2( 1113857χσ cosσπ2

+ k2α1 minus k1

A1I(χ) χ3σ sin3σπ2

minus α1 + k3( 1113857χ2σ sin σπ

+ α1k3 minus k2( 1113857χσ sinσπ2

A2R(χ) α1χ2σ cos σπ minus α1k3χσ cosσπ2

minus k2α1

A2I(χ) α1χ2σ sin σπ minus α1k3χσ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In view of (15) we have

cos χσ minusA1R(χ)A2R(χ) + A1I(χ)A2I(χ)

A22R(χ) + A2

2I(χ)

sin χσ minusA1R(χ)A2I(χ) minus A1I(χ)A2R(χ)

A22R(χ) + A2

2I(χ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

Let

p1 cos3σπ2

p2 minus α1 + k3( 1113857cos σπ

p3 α1k3 minus k2( 1113857cosσπ2

p4 k2α1 minus k1

p5 sin3σπ2

p6 minus α1 + k3( 1113857sin σπ

p7 α1k3 minus k2( 1113857sinσπ2

p8 α1 cos σπ

p9 minus α1k3 cosσπ2

p10 minus k2α1

p11 α1 sin σπ

p12 minus α1k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

HenceA1R(χ) p1χ3σ + p2χ2σ + p3χσ + p4

A1I(χ) p5χ3σ + p6χ2σ + p7χσ

A2R(χ) p8χ2σ + p9χσ + p10

A2I(χ) p11χ2σ + p12χσ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

By (19) we have

A1R(χ)A2R(χ) + A1I(χ)A2I(χ)1113858 11138592

+ A1R(χ)A2I(χ) minus A1I(χ)A2R(χ)1113858 11138592

A22R(χ) + A

22I(χ)1113960 1113961

2

(20)

In addition

A1R(χ)A2R(χ) + A1I(χ)A2I(χ)1113858 11138592

q1χ5σ

+ q2χ4σ

+ q3χ3σ

+ q4χ2σ

+ q5χσ

+ q61113872 11138732

A1R(χ)A2I(χ) minus A1I(χ)A2R(χ)1113858 11138592

q7χ5σ

+ b8χ4σ

+ q9χ3ϑ

+ q10χ2σ

+ q11χσ

1113872 11138732

A22R(χ) + A

22I(χ)1113960 1113961

2 q12χ

4q+ q13χ

3ϑ+ q14χ

2σ+ q15χ

σ1113872 1113873

2

(21)

whereq1 p1p8 + p5p11

q2 p2p8 + p1p9 + p6p11 + p5p12

q3 p3p8 + p2p9 + p1p10 + p7p11 + p6p12

q4 p4p8 + p3p9 + p2p10 + p7p12

q5 p4p9 + p3p10

q6 p4p10

q7 p1p11 minus p5p8

q8 p2p11 + p1p12 minus p5p9 minus p6p8

q9 p3p11 + p2p12 minus p5p10 minus p6p9 minus p7p8

q10 p4p11 + p3p12 minus p6p10 minus p7p9

q11 p4p12 minus p7p10

q12 p28 + p2

11

q13 2 p8p9 + p11p12( 1113857

q14 2p8p10 + p212

q15 2p9p10

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

By (20) we have

ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(23)

Complexity 5

whereξ1 q21 + q27

ξ2 2q1q2 + 2q7q8

ξ3 q22 + b28 minus q212 + 2q1q3 + 2q7q9

ξ4 2 q2q3 + q1q2 + q7q10 + q8q9 minus q12q13( 1113857

ξ5 q23 + 2q1q5 + 2q2q4 + q29 + 2q7q11 + 2q8q10 minus q213 minus 2q12q14

ξ6 2q1q6 + 2q2q5 + 2q3q4 + 2q9q10 minus 2q12q15 minus 2q13q14

ξ7 q24 + 2q3q5 + q210 + 2q8q11 + 2q9q11 minus q214 minus 2q13q15

ξ8 2q3q6 + 2q4q5 + 2q10q11 minus 2q14q15

ξ9 q25 + 2q4q6 + q211 minus q215

ξ10 2q5q6

ξ11 q26

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

Let

W(χ) ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(25)

V(μ) ξ1μ10

+ ξ2μ9

+ ξ3μ8

+ ξ4μ7

+ ξ5μ6

+ ξ6μ5

+ ξ7μ4

+ ξ8μ3

+ ξ9μ2

+ ξ10μ + ξ11(26)

Lemma 3

(i) Assume that ξj gt 0(j 1 2 11) then the root that haszero real parts does not exist in (13)(ii) Assume that ξ11 lt 0 and existμ0 gt 0 which satisfies V(μ0)lt 0then (13) possesses at least two pairs of purely imaginary roots

Proof of Lemma 3(i) According to (25) we havedW(χ)

dχ 10σξ1χ

10σminus 1+ 9σξ2χ

9σminus 1+ 8σξ3χ

8σminus 1+ 7σξ4χ

7σminus 1

+ 6σξ5χ6σminus 1

+ 5σξ6χ5σminus 1

+ 4σξ7χ4σminus 1

+ 3σξ8χ3σminus 1

+ 2σξ9χ2σminus 1

+ σξ10χσminus 1

(27)

In view of ξi gt 0(i 1 2 10) we have ((dW(χ))dχ)gt 0 forall χ gt 0 In view of W(0) ξ11 gt 0 we know that (26)does not possess positive real roots Furthermore s 0 is notthe root of (13) +is proves Lemma 3(i)

Proof of Lemma 3(ii) According to V(0) ξ11 gt0 V(ε0)lt 0(ε0 gt 0) and limτ⟶+infin(V(τ))dτ +infinexist ε01 isin (0 ε0) and ε02 isin (ε0 +infin) such that V(ε01)

V(ε02) 0 Hence (26) possesses at least two positive realroots and (13) possesses at least two pairs of purely imag-inary roots +is proves Lemma 3 (ii)

Suppose that (23) possesses ten positive real rootsζj(j 1 2 10) In view of (17) we get

9hm

1χm

arccos minusA1R χm( 1113857A2R χm( 1113857 + A1I χm( 1113857A2I χm( 1113857

A22R χm( 1113857 + A2

2I χm( 11138571113888 1113889 + 2hπ1113890 1113891

(28)

where h 0 1 2 andm 1 2 10 Hence plusmn iχm is apair of purely imaginary roots of (13) when 9 9h

m Let

90 minm1211

90m1113966 1113967

χ0 χ|990

(29)

Next the hypothesis H1 is given K1S1 + K2S2 gt 0

where

K1 3σχ3σminus 10 cos

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 cos

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 cos

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891cos χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891sin χ090

K2 3σχ3σminus 10 sin

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 sin

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 sin

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891sin χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891cos χ090

S1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

S2 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 cos χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 sin χ090

(30)

Lemma 4 Suppose that s(9) c(9) + iϕ(9) is the root of(13) around 9 90 such that c(90) 0 ϕ(90) ϕ0 thenRe[dsd9]|990 ϕϕ0 gt 0

Proof of Lemma 4 In view of (13) we get

dA1(s)

d9+

dA2(s)

d9e

minus s9minus e

minus s9 ds

d99 + s1113888 1113889A2(s) 0 (31)

6 Complexity

SincedA1(s)

d9 3σs

3σminus 1ds

d9minus 2σ α1 + k3( 1113857s

2σminus 1ds

d9+ σ α1k3 minus k2( 1113857s

σminus 1ds

d9

3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

1113960 1113961ds

d9

dA2(s)

d9 2α1σs

2σminus 1minus α1k3σs

σminus 11113872 1113873

ds

d9

(32)

we obtainds

d91113890 1113891

minus 1

P1(9)

P2(9)minus

9

s (33)

where

P1(s) 3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

+ 2α1σs2σminus 1

minus α1k3σsσminus 1

1113872 1113873eminus s9

P2(s) A2(s)seminus s9

(34)

+us

Reds

d91113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

ReP1(9)

P2(9)1113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

K1S1 + K2S2

S21 + S2

2

(35)

It follows from H1 that

Reds

d91113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ψψ0

gt 0 (36)

+is proves Lemma 4

Next the hypothesis H2 is given

k3 lt 0

k2k3 gt minus 2k1

k1 lt 0

(37)

Lemma 5 If 9 0 and H2 holds true then system (13) islocally asymptotically stable

Proof of Lemma 5 Under the condition 9 0 (13) becomes

λ3 minus k3λ2

minus k2λ minus k1 0 (38)

In view of H2 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (38) +us Lemma 5 holds true implying theproof

+e following results are established by above discussion

Theorem 1 (under the condition of Lemma 3(ii)) Supposethat H1 and H2 hold true then (i) the equilibrium point of

system (10) at the origin is locally asymptotically stable if0le 9lt 90 and (ii) a Hopf bifurcation will happen near theequilibrium point E1 if 9 90

32 Adding Time-Delayed Feedback Controller to the Sec-ond Equation of Model (2) In this section we add a time-delayed feedback controller α2[w2(t) minus w2(t minus ρ)] tothe second equation of model (2) and then system (2)becomes

dσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)1113858 1113859

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(39)

where α2 is the gain coefficient and ρ is the time delay +elinear equation of equation (39) around E1 (0 0 0) is

dσw1

dtσ w2

dσw2

dtσ α2w2 + w3 minus α2w2(t minus ρ)

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

+e characteristic equation of (40) takes the followingform

det

sσ minus 1 0

0 sσ minus α2 + α2eminus sρ minus 1

minus k11 minus k2 sρ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (41)

Hence

B1(s) + B2(s)eminus sρ

0 (42)

where

B1(s) s3σ

minus α2 + k3 + 1( 1113857s2σ

+ α2k3 + k3 minus k2( 1113857sσ

minus k1

B2(s) α2s2σ

minus α2k3sσ

(43)

Assume that s iϑ ϑ(cos(π2) + i sin(π2)) is the rootof (42) then we have

B2R(ϑ)cos ϑρ + B2I(ϑ)sin ζρ minus B1R(ϑ)

B2I(ϑ)cos ϑρ minus B2R(ϕ)sin ϑρ minus B1I(ϑ)1113896 (44)

where

Complexity 7

B1R(ϑ) ϑ3σ cos3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ cos σπ

+ α2k3 + k3 minus k2( 1113857ϑσ cosσπ2

minus k1

B1I(ϑ) ϑ3σ sin3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ sin σπ

+ α2k3 + k3 minus k2( 1113857ϑσ sinσπ2

B2R(ϑ) α2ϑ2ϑ cos ϑπ minus α2k3ϑ

σ cosσπ2

B2I(ϑ) α2ϑ2ϑ sin ϑπ minus α2k3ϑ

σ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

According to (44) we get

cos ϑρ minusB1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)

B22R(ϑ) + B2

2I(ϑ)

sin ϑρ minusB1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)

B22R(ϑ) + B2

2I(ϑ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

Let

s1 cos3σπ2

s2 minus α2 + k3 + 1( 1113857cos σπ

s3 α2k3 + k3 minus k2( 1113857cosσπ2

s4 minus k1

s5 sin3σπ2

s6 minus α2 + k3 + 1( 1113857sin σπ

s7 α2k3 + k3 minus k2( 1113857sinσπ2

s8 α2 cos ϑπ

s9 minus α2k3 cosσπ2

s10 α2 sin ϑπ

s11 minus α2k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

+en

B1R(ϑ) s1ϑ3σ + s2ϑ

2σ + s3ϑσ + s4

B1I(ϑ) s5ϑ3σ + s6ϑ

2σ + s7ϑσ

B2R(ϑ) s8ϑ2σ + s9ϑ

σ

B2I(ϑ) s10ϑ2σ + s11ϑ

σ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(48)

By (46) we have

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

+ B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

B22R(ϑ) + B

22I(ϑ)1113960 1113961

2

(49)

Furthermore

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

t1ϑ5σ

+ t2ϑ4σ

+ t3ϑ3σ

+ t4ϑ2σ

+ t5ϑσ

1113872 11138732

B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

t5ϑ5σ

+ t6ϑ4σ

+ t7ϑ3ϑ

+ t8ϑ2σ

+ t9ϑσ

1113872 11138732

B22R(ϑ) + B

22I(χ)1113960 1113961

2 t10ϑ

4σ+ t11ϑ

3σ+ t12ϑ

2σ+ t13ϑ

σ1113872 1113873

2

(50)

wheret1 s1s8 + s5s10

t2 s2s8 + s1s9 + s6s10 + s5s11

t3 s3s8 + s2s9 + s7s10 + s6s11

t4 s4s8 + s3s9 + s7s11

t5 s4s9

t6 s1s10 minus s5s8

t7 s2s10 + s1s11 minus s5s9 minus s6s8

t8 s3s10 + s2s11 minus s6s9 minus s7s8

t9 s4s10 + s3s11 minus s7s9

t10 s4s11

t11 s28 + s210

t12 2 s8s9 + s10s11( 1113857

t13 s211

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(51)

In view of (49) we have

κ1ϑ10σ

+ κ2ϑ9σ

+ κ3ϑ8σ

+ κ4ϑ7σ

+ κ5ϑ6σ

+ κ6ϑ5σ

+ κ7ϑ4σ

+ κ8ϑ3σ

+ κ9ϑ2σ

0

(52)

where

8 Complexity

κ1 t21 + t26

κ2 2t1t2 + 2t6t7

κ3 t22 + t27 minus t211 + 2t1t3 + 2t6t8

κ4 2 t2t3 + t1t2 + t6t9 + t7t8 minus t11t12( 1113857

κ5 t23 + 2t2t4 + t28 + 2t6t10 + 2t7t9 minus t212 minus 2t11t13

κ6 2t3t4 + 2t8t9

κ7 t24 + t29 + 2t7t10 + 2t8t10 minus t213

κ8 2t9t10 + 2t4t5

κ9 t25 + t210

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(53)

By (52) we have

κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9 0

(54)

Let

K(ϑ) κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9(55)

M(ζ) κ1ζ8

+ κ2ζ7

+ κ3ζ6

+ κ4ζ5

+ κ5ζ4

+ κ6ζ3

+ κ7ζ2

+ κ8ζ1

+ κ9(56)

Lemma 6

(i) Suppose that κj gt 0(j 1 2 9) then the root that haszero real parts does not exist in (42)(ii) Suppose that κ9 lt 0 and existη0 gt 0 which satisfies M(η0)lt 0then (42) possesses at least two pairs of purely imaginary roots

Proof of Lemma 6(i) In view of (55) we havedK(ϑ)

dχ 8κ1σϑ

8σminus 1+ 7κ2σϑ

7σminus 1+ 6κ3σϑ

6σminus 1+ 5κ4σϑ

5σminus 1

+ 4κ5σϑ4σminus 1

+ 3κ6σϑ3σminus 1

+ 2κ7σϑ2σminus 1

+ κ8σϑσminus 1

(57)

In view of κi gt 0(i 1 2 9) we have((dK(ϑ))dϑ) gt 0 forall ϑgt 0 In view of K(0) κ9 gt 0 we knowthat (56) does not possess positive real roots Furthermores 0 is not the root of (42) +is proves Lemma 6(i)

Proof of Lemma 6(ii) In view of M(0) κ9 gt 0 M(η0)lt0(η0 gt 0) and limε⟶+infin((M(ε))dε) +infin exist η01 isin (0 η0)and η02 isin (η0 +infin) such that M(η01) M(η02) 0 Hence(55) has at least two positive real roots and (42) has at leasttwo pairs of purely imaginary roots +is proves Lemma6(ii)

Assume that (54) has eight positive real rootsϑj(j 1 2 8) By (46) we have

ριk 1ϑk

arccos minusB1R ϑm( 1113857B2R ϑk( 1113857 + B1I ϑk( 1113857B2I ϑk( 1113857

B22R ϑk( 1113857 + B2

2I ϑk( 11138571113888 1113889 + 2ιπ1113890 1113891

(58)

where ι 0 1 2 and k 1 2 8 Hence plusmn iϑk is apair of purely imaginary roots of (42) when ρ ριk Let

ρ0 mink128

ρ0k1113966 1113967

ϑ0 ϑ|ρρ0(59)

Next we give the hypothesis H3 as followsF1G1 + F2G2 gt 0 where

F1 3σϑ3σminus 10 cos

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 cos

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 cos

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

F2 3σϑ3σminus 10 sin

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 sin

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 sin

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

G1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

G2 α2ϑ2σ0 cos σπ minus α2k3χ

σ0 cos

σπ2

minus α2k31113876 1113877χ0 cos ϑ0ρ0

+ α2ϑ2σ0 sin σπ minus α2k3ϑ

σ0 sin

σπ2

minus α2k31113876 1113877ϑ0 sin ϑ0ρ0

(60)

Lemma 7 Assume that s(ρ) ζ(ρ) + iϖ(ρ) is the root of(42) around ρ ρ0 such that ζ(ρ0) 0 ϖ(ρ0) ϖ0 thenRe[dsdρ]|ρρ0 ϖϖ0 gt 0

Proof of Lemma 7 By (42) we have

dB1(s)

dρ+

dB2(s)

dρe

minus sρminus e

minus sρ ds

dρρ + s1113888 1113889B2(s) 0 (61)

Complexity 9

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

A1R(χ) χ3σ cos3σπ2

minus α1 + k3( 1113857χ2σ cos σπ

+ α1k3 minus k2( 1113857χσ cosσπ2

+ k2α1 minus k1

A1I(χ) χ3σ sin3σπ2

minus α1 + k3( 1113857χ2σ sin σπ

+ α1k3 minus k2( 1113857χσ sinσπ2

A2R(χ) α1χ2σ cos σπ minus α1k3χσ cosσπ2

minus k2α1

A2I(χ) α1χ2σ sin σπ minus α1k3χσ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

In view of (15) we have

cos χσ minusA1R(χ)A2R(χ) + A1I(χ)A2I(χ)

A22R(χ) + A2

2I(χ)

sin χσ minusA1R(χ)A2I(χ) minus A1I(χ)A2R(χ)

A22R(χ) + A2

2I(χ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

Let

p1 cos3σπ2

p2 minus α1 + k3( 1113857cos σπ

p3 α1k3 minus k2( 1113857cosσπ2

p4 k2α1 minus k1

p5 sin3σπ2

p6 minus α1 + k3( 1113857sin σπ

p7 α1k3 minus k2( 1113857sinσπ2

p8 α1 cos σπ

p9 minus α1k3 cosσπ2

p10 minus k2α1

p11 α1 sin σπ

p12 minus α1k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

HenceA1R(χ) p1χ3σ + p2χ2σ + p3χσ + p4

A1I(χ) p5χ3σ + p6χ2σ + p7χσ

A2R(χ) p8χ2σ + p9χσ + p10

A2I(χ) p11χ2σ + p12χσ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

By (19) we have

A1R(χ)A2R(χ) + A1I(χ)A2I(χ)1113858 11138592

+ A1R(χ)A2I(χ) minus A1I(χ)A2R(χ)1113858 11138592

A22R(χ) + A

22I(χ)1113960 1113961

2

(20)

In addition

A1R(χ)A2R(χ) + A1I(χ)A2I(χ)1113858 11138592

q1χ5σ

+ q2χ4σ

+ q3χ3σ

+ q4χ2σ

+ q5χσ

+ q61113872 11138732

A1R(χ)A2I(χ) minus A1I(χ)A2R(χ)1113858 11138592

q7χ5σ

+ b8χ4σ

+ q9χ3ϑ

+ q10χ2σ

+ q11χσ

1113872 11138732

A22R(χ) + A

22I(χ)1113960 1113961

2 q12χ

4q+ q13χ

3ϑ+ q14χ

2σ+ q15χ

σ1113872 1113873

2

(21)

whereq1 p1p8 + p5p11

q2 p2p8 + p1p9 + p6p11 + p5p12

q3 p3p8 + p2p9 + p1p10 + p7p11 + p6p12

q4 p4p8 + p3p9 + p2p10 + p7p12

q5 p4p9 + p3p10

q6 p4p10

q7 p1p11 minus p5p8

q8 p2p11 + p1p12 minus p5p9 minus p6p8

q9 p3p11 + p2p12 minus p5p10 minus p6p9 minus p7p8

q10 p4p11 + p3p12 minus p6p10 minus p7p9

q11 p4p12 minus p7p10

q12 p28 + p2

11

q13 2 p8p9 + p11p12( 1113857

q14 2p8p10 + p212

q15 2p9p10

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

By (20) we have

ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(23)

Complexity 5

whereξ1 q21 + q27

ξ2 2q1q2 + 2q7q8

ξ3 q22 + b28 minus q212 + 2q1q3 + 2q7q9

ξ4 2 q2q3 + q1q2 + q7q10 + q8q9 minus q12q13( 1113857

ξ5 q23 + 2q1q5 + 2q2q4 + q29 + 2q7q11 + 2q8q10 minus q213 minus 2q12q14

ξ6 2q1q6 + 2q2q5 + 2q3q4 + 2q9q10 minus 2q12q15 minus 2q13q14

ξ7 q24 + 2q3q5 + q210 + 2q8q11 + 2q9q11 minus q214 minus 2q13q15

ξ8 2q3q6 + 2q4q5 + 2q10q11 minus 2q14q15

ξ9 q25 + 2q4q6 + q211 minus q215

ξ10 2q5q6

ξ11 q26

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

Let

W(χ) ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(25)

V(μ) ξ1μ10

+ ξ2μ9

+ ξ3μ8

+ ξ4μ7

+ ξ5μ6

+ ξ6μ5

+ ξ7μ4

+ ξ8μ3

+ ξ9μ2

+ ξ10μ + ξ11(26)

Lemma 3

(i) Assume that ξj gt 0(j 1 2 11) then the root that haszero real parts does not exist in (13)(ii) Assume that ξ11 lt 0 and existμ0 gt 0 which satisfies V(μ0)lt 0then (13) possesses at least two pairs of purely imaginary roots

Proof of Lemma 3(i) According to (25) we havedW(χ)

dχ 10σξ1χ

10σminus 1+ 9σξ2χ

9σminus 1+ 8σξ3χ

8σminus 1+ 7σξ4χ

7σminus 1

+ 6σξ5χ6σminus 1

+ 5σξ6χ5σminus 1

+ 4σξ7χ4σminus 1

+ 3σξ8χ3σminus 1

+ 2σξ9χ2σminus 1

+ σξ10χσminus 1

(27)

In view of ξi gt 0(i 1 2 10) we have ((dW(χ))dχ)gt 0 forall χ gt 0 In view of W(0) ξ11 gt 0 we know that (26)does not possess positive real roots Furthermore s 0 is notthe root of (13) +is proves Lemma 3(i)

Proof of Lemma 3(ii) According to V(0) ξ11 gt0 V(ε0)lt 0(ε0 gt 0) and limτ⟶+infin(V(τ))dτ +infinexist ε01 isin (0 ε0) and ε02 isin (ε0 +infin) such that V(ε01)

V(ε02) 0 Hence (26) possesses at least two positive realroots and (13) possesses at least two pairs of purely imag-inary roots +is proves Lemma 3 (ii)

Suppose that (23) possesses ten positive real rootsζj(j 1 2 10) In view of (17) we get

9hm

1χm

arccos minusA1R χm( 1113857A2R χm( 1113857 + A1I χm( 1113857A2I χm( 1113857

A22R χm( 1113857 + A2

2I χm( 11138571113888 1113889 + 2hπ1113890 1113891

(28)

where h 0 1 2 andm 1 2 10 Hence plusmn iχm is apair of purely imaginary roots of (13) when 9 9h

m Let

90 minm1211

90m1113966 1113967

χ0 χ|990

(29)

Next the hypothesis H1 is given K1S1 + K2S2 gt 0

where

K1 3σχ3σminus 10 cos

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 cos

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 cos

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891cos χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891sin χ090

K2 3σχ3σminus 10 sin

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 sin

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 sin

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891sin χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891cos χ090

S1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

S2 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 cos χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 sin χ090

(30)

Lemma 4 Suppose that s(9) c(9) + iϕ(9) is the root of(13) around 9 90 such that c(90) 0 ϕ(90) ϕ0 thenRe[dsd9]|990 ϕϕ0 gt 0

Proof of Lemma 4 In view of (13) we get

dA1(s)

d9+

dA2(s)

d9e

minus s9minus e

minus s9 ds

d99 + s1113888 1113889A2(s) 0 (31)

6 Complexity

SincedA1(s)

d9 3σs

3σminus 1ds

d9minus 2σ α1 + k3( 1113857s

2σminus 1ds

d9+ σ α1k3 minus k2( 1113857s

σminus 1ds

d9

3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

1113960 1113961ds

d9

dA2(s)

d9 2α1σs

2σminus 1minus α1k3σs

σminus 11113872 1113873

ds

d9

(32)

we obtainds

d91113890 1113891

minus 1

P1(9)

P2(9)minus

9

s (33)

where

P1(s) 3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

+ 2α1σs2σminus 1

minus α1k3σsσminus 1

1113872 1113873eminus s9

P2(s) A2(s)seminus s9

(34)

+us

Reds

d91113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

ReP1(9)

P2(9)1113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

K1S1 + K2S2

S21 + S2

2

(35)

It follows from H1 that

Reds

d91113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ψψ0

gt 0 (36)

+is proves Lemma 4

Next the hypothesis H2 is given

k3 lt 0

k2k3 gt minus 2k1

k1 lt 0

(37)

Lemma 5 If 9 0 and H2 holds true then system (13) islocally asymptotically stable

Proof of Lemma 5 Under the condition 9 0 (13) becomes

λ3 minus k3λ2

minus k2λ minus k1 0 (38)

In view of H2 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (38) +us Lemma 5 holds true implying theproof

+e following results are established by above discussion

Theorem 1 (under the condition of Lemma 3(ii)) Supposethat H1 and H2 hold true then (i) the equilibrium point of

system (10) at the origin is locally asymptotically stable if0le 9lt 90 and (ii) a Hopf bifurcation will happen near theequilibrium point E1 if 9 90

32 Adding Time-Delayed Feedback Controller to the Sec-ond Equation of Model (2) In this section we add a time-delayed feedback controller α2[w2(t) minus w2(t minus ρ)] tothe second equation of model (2) and then system (2)becomes

dσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)1113858 1113859

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(39)

where α2 is the gain coefficient and ρ is the time delay +elinear equation of equation (39) around E1 (0 0 0) is

dσw1

dtσ w2

dσw2

dtσ α2w2 + w3 minus α2w2(t minus ρ)

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

+e characteristic equation of (40) takes the followingform

det

sσ minus 1 0

0 sσ minus α2 + α2eminus sρ minus 1

minus k11 minus k2 sρ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (41)

Hence

B1(s) + B2(s)eminus sρ

0 (42)

where

B1(s) s3σ

minus α2 + k3 + 1( 1113857s2σ

+ α2k3 + k3 minus k2( 1113857sσ

minus k1

B2(s) α2s2σ

minus α2k3sσ

(43)

Assume that s iϑ ϑ(cos(π2) + i sin(π2)) is the rootof (42) then we have

B2R(ϑ)cos ϑρ + B2I(ϑ)sin ζρ minus B1R(ϑ)

B2I(ϑ)cos ϑρ minus B2R(ϕ)sin ϑρ minus B1I(ϑ)1113896 (44)

where

Complexity 7

B1R(ϑ) ϑ3σ cos3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ cos σπ

+ α2k3 + k3 minus k2( 1113857ϑσ cosσπ2

minus k1

B1I(ϑ) ϑ3σ sin3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ sin σπ

+ α2k3 + k3 minus k2( 1113857ϑσ sinσπ2

B2R(ϑ) α2ϑ2ϑ cos ϑπ minus α2k3ϑ

σ cosσπ2

B2I(ϑ) α2ϑ2ϑ sin ϑπ minus α2k3ϑ

σ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

According to (44) we get

cos ϑρ minusB1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)

B22R(ϑ) + B2

2I(ϑ)

sin ϑρ minusB1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)

B22R(ϑ) + B2

2I(ϑ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

Let

s1 cos3σπ2

s2 minus α2 + k3 + 1( 1113857cos σπ

s3 α2k3 + k3 minus k2( 1113857cosσπ2

s4 minus k1

s5 sin3σπ2

s6 minus α2 + k3 + 1( 1113857sin σπ

s7 α2k3 + k3 minus k2( 1113857sinσπ2

s8 α2 cos ϑπ

s9 minus α2k3 cosσπ2

s10 α2 sin ϑπ

s11 minus α2k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

+en

B1R(ϑ) s1ϑ3σ + s2ϑ

2σ + s3ϑσ + s4

B1I(ϑ) s5ϑ3σ + s6ϑ

2σ + s7ϑσ

B2R(ϑ) s8ϑ2σ + s9ϑ

σ

B2I(ϑ) s10ϑ2σ + s11ϑ

σ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(48)

By (46) we have

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

+ B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

B22R(ϑ) + B

22I(ϑ)1113960 1113961

2

(49)

Furthermore

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

t1ϑ5σ

+ t2ϑ4σ

+ t3ϑ3σ

+ t4ϑ2σ

+ t5ϑσ

1113872 11138732

B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

t5ϑ5σ

+ t6ϑ4σ

+ t7ϑ3ϑ

+ t8ϑ2σ

+ t9ϑσ

1113872 11138732

B22R(ϑ) + B

22I(χ)1113960 1113961

2 t10ϑ

4σ+ t11ϑ

3σ+ t12ϑ

2σ+ t13ϑ

σ1113872 1113873

2

(50)

wheret1 s1s8 + s5s10

t2 s2s8 + s1s9 + s6s10 + s5s11

t3 s3s8 + s2s9 + s7s10 + s6s11

t4 s4s8 + s3s9 + s7s11

t5 s4s9

t6 s1s10 minus s5s8

t7 s2s10 + s1s11 minus s5s9 minus s6s8

t8 s3s10 + s2s11 minus s6s9 minus s7s8

t9 s4s10 + s3s11 minus s7s9

t10 s4s11

t11 s28 + s210

t12 2 s8s9 + s10s11( 1113857

t13 s211

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(51)

In view of (49) we have

κ1ϑ10σ

+ κ2ϑ9σ

+ κ3ϑ8σ

+ κ4ϑ7σ

+ κ5ϑ6σ

+ κ6ϑ5σ

+ κ7ϑ4σ

+ κ8ϑ3σ

+ κ9ϑ2σ

0

(52)

where

8 Complexity

κ1 t21 + t26

κ2 2t1t2 + 2t6t7

κ3 t22 + t27 minus t211 + 2t1t3 + 2t6t8

κ4 2 t2t3 + t1t2 + t6t9 + t7t8 minus t11t12( 1113857

κ5 t23 + 2t2t4 + t28 + 2t6t10 + 2t7t9 minus t212 minus 2t11t13

κ6 2t3t4 + 2t8t9

κ7 t24 + t29 + 2t7t10 + 2t8t10 minus t213

κ8 2t9t10 + 2t4t5

κ9 t25 + t210

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(53)

By (52) we have

κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9 0

(54)

Let

K(ϑ) κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9(55)

M(ζ) κ1ζ8

+ κ2ζ7

+ κ3ζ6

+ κ4ζ5

+ κ5ζ4

+ κ6ζ3

+ κ7ζ2

+ κ8ζ1

+ κ9(56)

Lemma 6

(i) Suppose that κj gt 0(j 1 2 9) then the root that haszero real parts does not exist in (42)(ii) Suppose that κ9 lt 0 and existη0 gt 0 which satisfies M(η0)lt 0then (42) possesses at least two pairs of purely imaginary roots

Proof of Lemma 6(i) In view of (55) we havedK(ϑ)

dχ 8κ1σϑ

8σminus 1+ 7κ2σϑ

7σminus 1+ 6κ3σϑ

6σminus 1+ 5κ4σϑ

5σminus 1

+ 4κ5σϑ4σminus 1

+ 3κ6σϑ3σminus 1

+ 2κ7σϑ2σminus 1

+ κ8σϑσminus 1

(57)

In view of κi gt 0(i 1 2 9) we have((dK(ϑ))dϑ) gt 0 forall ϑgt 0 In view of K(0) κ9 gt 0 we knowthat (56) does not possess positive real roots Furthermores 0 is not the root of (42) +is proves Lemma 6(i)

Proof of Lemma 6(ii) In view of M(0) κ9 gt 0 M(η0)lt0(η0 gt 0) and limε⟶+infin((M(ε))dε) +infin exist η01 isin (0 η0)and η02 isin (η0 +infin) such that M(η01) M(η02) 0 Hence(55) has at least two positive real roots and (42) has at leasttwo pairs of purely imaginary roots +is proves Lemma6(ii)

Assume that (54) has eight positive real rootsϑj(j 1 2 8) By (46) we have

ριk 1ϑk

arccos minusB1R ϑm( 1113857B2R ϑk( 1113857 + B1I ϑk( 1113857B2I ϑk( 1113857

B22R ϑk( 1113857 + B2

2I ϑk( 11138571113888 1113889 + 2ιπ1113890 1113891

(58)

where ι 0 1 2 and k 1 2 8 Hence plusmn iϑk is apair of purely imaginary roots of (42) when ρ ριk Let

ρ0 mink128

ρ0k1113966 1113967

ϑ0 ϑ|ρρ0(59)

Next we give the hypothesis H3 as followsF1G1 + F2G2 gt 0 where

F1 3σϑ3σminus 10 cos

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 cos

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 cos

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

F2 3σϑ3σminus 10 sin

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 sin

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 sin

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

G1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

G2 α2ϑ2σ0 cos σπ minus α2k3χ

σ0 cos

σπ2

minus α2k31113876 1113877χ0 cos ϑ0ρ0

+ α2ϑ2σ0 sin σπ minus α2k3ϑ

σ0 sin

σπ2

minus α2k31113876 1113877ϑ0 sin ϑ0ρ0

(60)

Lemma 7 Assume that s(ρ) ζ(ρ) + iϖ(ρ) is the root of(42) around ρ ρ0 such that ζ(ρ0) 0 ϖ(ρ0) ϖ0 thenRe[dsdρ]|ρρ0 ϖϖ0 gt 0

Proof of Lemma 7 By (42) we have

dB1(s)

dρ+

dB2(s)

dρe

minus sρminus e

minus sρ ds

dρρ + s1113888 1113889B2(s) 0 (61)

Complexity 9

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

whereξ1 q21 + q27

ξ2 2q1q2 + 2q7q8

ξ3 q22 + b28 minus q212 + 2q1q3 + 2q7q9

ξ4 2 q2q3 + q1q2 + q7q10 + q8q9 minus q12q13( 1113857

ξ5 q23 + 2q1q5 + 2q2q4 + q29 + 2q7q11 + 2q8q10 minus q213 minus 2q12q14

ξ6 2q1q6 + 2q2q5 + 2q3q4 + 2q9q10 minus 2q12q15 minus 2q13q14

ξ7 q24 + 2q3q5 + q210 + 2q8q11 + 2q9q11 minus q214 minus 2q13q15

ξ8 2q3q6 + 2q4q5 + 2q10q11 minus 2q14q15

ξ9 q25 + 2q4q6 + q211 minus q215

ξ10 2q5q6

ξ11 q26

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(24)

Let

W(χ) ξ1χ10σ

+ ξ2χ9σ

+ ξ3χ8σ

+ ξ4χ7σ

+ ξ5χ6σ

+ ξ6χ5σ

+ ξ7χ4σ

+ ξ8χ3σ

+ ξ9χ2σ

+ ξ10χσ

+ ξ11 0

(25)

V(μ) ξ1μ10

+ ξ2μ9

+ ξ3μ8

+ ξ4μ7

+ ξ5μ6

+ ξ6μ5

+ ξ7μ4

+ ξ8μ3

+ ξ9μ2

+ ξ10μ + ξ11(26)

Lemma 3

(i) Assume that ξj gt 0(j 1 2 11) then the root that haszero real parts does not exist in (13)(ii) Assume that ξ11 lt 0 and existμ0 gt 0 which satisfies V(μ0)lt 0then (13) possesses at least two pairs of purely imaginary roots

Proof of Lemma 3(i) According to (25) we havedW(χ)

dχ 10σξ1χ

10σminus 1+ 9σξ2χ

9σminus 1+ 8σξ3χ

8σminus 1+ 7σξ4χ

7σminus 1

+ 6σξ5χ6σminus 1

+ 5σξ6χ5σminus 1

+ 4σξ7χ4σminus 1

+ 3σξ8χ3σminus 1

+ 2σξ9χ2σminus 1

+ σξ10χσminus 1

(27)

In view of ξi gt 0(i 1 2 10) we have ((dW(χ))dχ)gt 0 forall χ gt 0 In view of W(0) ξ11 gt 0 we know that (26)does not possess positive real roots Furthermore s 0 is notthe root of (13) +is proves Lemma 3(i)

Proof of Lemma 3(ii) According to V(0) ξ11 gt0 V(ε0)lt 0(ε0 gt 0) and limτ⟶+infin(V(τ))dτ +infinexist ε01 isin (0 ε0) and ε02 isin (ε0 +infin) such that V(ε01)

V(ε02) 0 Hence (26) possesses at least two positive realroots and (13) possesses at least two pairs of purely imag-inary roots +is proves Lemma 3 (ii)

Suppose that (23) possesses ten positive real rootsζj(j 1 2 10) In view of (17) we get

9hm

1χm

arccos minusA1R χm( 1113857A2R χm( 1113857 + A1I χm( 1113857A2I χm( 1113857

A22R χm( 1113857 + A2

2I χm( 11138571113888 1113889 + 2hπ1113890 1113891

(28)

where h 0 1 2 andm 1 2 10 Hence plusmn iχm is apair of purely imaginary roots of (13) when 9 9h

m Let

90 minm1211

90m1113966 1113967

χ0 χ|990

(29)

Next the hypothesis H1 is given K1S1 + K2S2 gt 0

where

K1 3σχ3σminus 10 cos

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 cos

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 cos

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891cos χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891sin χ090

K2 3σχ3σminus 10 sin

(3σ minus 1)π2

+ 2σ α1 + k3( 1113857χ2σminus 10 sin

(2σ minus 1)π2

+ σ α1k3 minus k2( 1113857χσminus 10 sin

(σ minus 1)π2

+ 2σα1χ2σminus 10 cos

(2σ minus 1)π2

minus σα1k3χσminus 10 cos

(σ minus 1)π2

1113890 1113891sin χ090

+ 2σα1χ2σminus 10 sin

(2σ minus 1)π2

minus σα1k3χσminus 10 sin

(σ minus 1)π2

1113890 1113891cos χ090

S1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

S2 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 cos χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 sin χ090

(30)

Lemma 4 Suppose that s(9) c(9) + iϕ(9) is the root of(13) around 9 90 such that c(90) 0 ϕ(90) ϕ0 thenRe[dsd9]|990 ϕϕ0 gt 0

Proof of Lemma 4 In view of (13) we get

dA1(s)

d9+

dA2(s)

d9e

minus s9minus e

minus s9 ds

d99 + s1113888 1113889A2(s) 0 (31)

6 Complexity

SincedA1(s)

d9 3σs

3σminus 1ds

d9minus 2σ α1 + k3( 1113857s

2σminus 1ds

d9+ σ α1k3 minus k2( 1113857s

σminus 1ds

d9

3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

1113960 1113961ds

d9

dA2(s)

d9 2α1σs

2σminus 1minus α1k3σs

σminus 11113872 1113873

ds

d9

(32)

we obtainds

d91113890 1113891

minus 1

P1(9)

P2(9)minus

9

s (33)

where

P1(s) 3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

+ 2α1σs2σminus 1

minus α1k3σsσminus 1

1113872 1113873eminus s9

P2(s) A2(s)seminus s9

(34)

+us

Reds

d91113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

ReP1(9)

P2(9)1113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

K1S1 + K2S2

S21 + S2

2

(35)

It follows from H1 that

Reds

d91113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ψψ0

gt 0 (36)

+is proves Lemma 4

Next the hypothesis H2 is given

k3 lt 0

k2k3 gt minus 2k1

k1 lt 0

(37)

Lemma 5 If 9 0 and H2 holds true then system (13) islocally asymptotically stable

Proof of Lemma 5 Under the condition 9 0 (13) becomes

λ3 minus k3λ2

minus k2λ minus k1 0 (38)

In view of H2 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (38) +us Lemma 5 holds true implying theproof

+e following results are established by above discussion

Theorem 1 (under the condition of Lemma 3(ii)) Supposethat H1 and H2 hold true then (i) the equilibrium point of

system (10) at the origin is locally asymptotically stable if0le 9lt 90 and (ii) a Hopf bifurcation will happen near theequilibrium point E1 if 9 90

32 Adding Time-Delayed Feedback Controller to the Sec-ond Equation of Model (2) In this section we add a time-delayed feedback controller α2[w2(t) minus w2(t minus ρ)] tothe second equation of model (2) and then system (2)becomes

dσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)1113858 1113859

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(39)

where α2 is the gain coefficient and ρ is the time delay +elinear equation of equation (39) around E1 (0 0 0) is

dσw1

dtσ w2

dσw2

dtσ α2w2 + w3 minus α2w2(t minus ρ)

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

+e characteristic equation of (40) takes the followingform

det

sσ minus 1 0

0 sσ minus α2 + α2eminus sρ minus 1

minus k11 minus k2 sρ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (41)

Hence

B1(s) + B2(s)eminus sρ

0 (42)

where

B1(s) s3σ

minus α2 + k3 + 1( 1113857s2σ

+ α2k3 + k3 minus k2( 1113857sσ

minus k1

B2(s) α2s2σ

minus α2k3sσ

(43)

Assume that s iϑ ϑ(cos(π2) + i sin(π2)) is the rootof (42) then we have

B2R(ϑ)cos ϑρ + B2I(ϑ)sin ζρ minus B1R(ϑ)

B2I(ϑ)cos ϑρ minus B2R(ϕ)sin ϑρ minus B1I(ϑ)1113896 (44)

where

Complexity 7

B1R(ϑ) ϑ3σ cos3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ cos σπ

+ α2k3 + k3 minus k2( 1113857ϑσ cosσπ2

minus k1

B1I(ϑ) ϑ3σ sin3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ sin σπ

+ α2k3 + k3 minus k2( 1113857ϑσ sinσπ2

B2R(ϑ) α2ϑ2ϑ cos ϑπ minus α2k3ϑ

σ cosσπ2

B2I(ϑ) α2ϑ2ϑ sin ϑπ minus α2k3ϑ

σ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

According to (44) we get

cos ϑρ minusB1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)

B22R(ϑ) + B2

2I(ϑ)

sin ϑρ minusB1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)

B22R(ϑ) + B2

2I(ϑ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

Let

s1 cos3σπ2

s2 minus α2 + k3 + 1( 1113857cos σπ

s3 α2k3 + k3 minus k2( 1113857cosσπ2

s4 minus k1

s5 sin3σπ2

s6 minus α2 + k3 + 1( 1113857sin σπ

s7 α2k3 + k3 minus k2( 1113857sinσπ2

s8 α2 cos ϑπ

s9 minus α2k3 cosσπ2

s10 α2 sin ϑπ

s11 minus α2k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

+en

B1R(ϑ) s1ϑ3σ + s2ϑ

2σ + s3ϑσ + s4

B1I(ϑ) s5ϑ3σ + s6ϑ

2σ + s7ϑσ

B2R(ϑ) s8ϑ2σ + s9ϑ

σ

B2I(ϑ) s10ϑ2σ + s11ϑ

σ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(48)

By (46) we have

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

+ B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

B22R(ϑ) + B

22I(ϑ)1113960 1113961

2

(49)

Furthermore

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

t1ϑ5σ

+ t2ϑ4σ

+ t3ϑ3σ

+ t4ϑ2σ

+ t5ϑσ

1113872 11138732

B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

t5ϑ5σ

+ t6ϑ4σ

+ t7ϑ3ϑ

+ t8ϑ2σ

+ t9ϑσ

1113872 11138732

B22R(ϑ) + B

22I(χ)1113960 1113961

2 t10ϑ

4σ+ t11ϑ

3σ+ t12ϑ

2σ+ t13ϑ

σ1113872 1113873

2

(50)

wheret1 s1s8 + s5s10

t2 s2s8 + s1s9 + s6s10 + s5s11

t3 s3s8 + s2s9 + s7s10 + s6s11

t4 s4s8 + s3s9 + s7s11

t5 s4s9

t6 s1s10 minus s5s8

t7 s2s10 + s1s11 minus s5s9 minus s6s8

t8 s3s10 + s2s11 minus s6s9 minus s7s8

t9 s4s10 + s3s11 minus s7s9

t10 s4s11

t11 s28 + s210

t12 2 s8s9 + s10s11( 1113857

t13 s211

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(51)

In view of (49) we have

κ1ϑ10σ

+ κ2ϑ9σ

+ κ3ϑ8σ

+ κ4ϑ7σ

+ κ5ϑ6σ

+ κ6ϑ5σ

+ κ7ϑ4σ

+ κ8ϑ3σ

+ κ9ϑ2σ

0

(52)

where

8 Complexity

κ1 t21 + t26

κ2 2t1t2 + 2t6t7

κ3 t22 + t27 minus t211 + 2t1t3 + 2t6t8

κ4 2 t2t3 + t1t2 + t6t9 + t7t8 minus t11t12( 1113857

κ5 t23 + 2t2t4 + t28 + 2t6t10 + 2t7t9 minus t212 minus 2t11t13

κ6 2t3t4 + 2t8t9

κ7 t24 + t29 + 2t7t10 + 2t8t10 minus t213

κ8 2t9t10 + 2t4t5

κ9 t25 + t210

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(53)

By (52) we have

κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9 0

(54)

Let

K(ϑ) κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9(55)

M(ζ) κ1ζ8

+ κ2ζ7

+ κ3ζ6

+ κ4ζ5

+ κ5ζ4

+ κ6ζ3

+ κ7ζ2

+ κ8ζ1

+ κ9(56)

Lemma 6

(i) Suppose that κj gt 0(j 1 2 9) then the root that haszero real parts does not exist in (42)(ii) Suppose that κ9 lt 0 and existη0 gt 0 which satisfies M(η0)lt 0then (42) possesses at least two pairs of purely imaginary roots

Proof of Lemma 6(i) In view of (55) we havedK(ϑ)

dχ 8κ1σϑ

8σminus 1+ 7κ2σϑ

7σminus 1+ 6κ3σϑ

6σminus 1+ 5κ4σϑ

5σminus 1

+ 4κ5σϑ4σminus 1

+ 3κ6σϑ3σminus 1

+ 2κ7σϑ2σminus 1

+ κ8σϑσminus 1

(57)

In view of κi gt 0(i 1 2 9) we have((dK(ϑ))dϑ) gt 0 forall ϑgt 0 In view of K(0) κ9 gt 0 we knowthat (56) does not possess positive real roots Furthermores 0 is not the root of (42) +is proves Lemma 6(i)

Proof of Lemma 6(ii) In view of M(0) κ9 gt 0 M(η0)lt0(η0 gt 0) and limε⟶+infin((M(ε))dε) +infin exist η01 isin (0 η0)and η02 isin (η0 +infin) such that M(η01) M(η02) 0 Hence(55) has at least two positive real roots and (42) has at leasttwo pairs of purely imaginary roots +is proves Lemma6(ii)

Assume that (54) has eight positive real rootsϑj(j 1 2 8) By (46) we have

ριk 1ϑk

arccos minusB1R ϑm( 1113857B2R ϑk( 1113857 + B1I ϑk( 1113857B2I ϑk( 1113857

B22R ϑk( 1113857 + B2

2I ϑk( 11138571113888 1113889 + 2ιπ1113890 1113891

(58)

where ι 0 1 2 and k 1 2 8 Hence plusmn iϑk is apair of purely imaginary roots of (42) when ρ ριk Let

ρ0 mink128

ρ0k1113966 1113967

ϑ0 ϑ|ρρ0(59)

Next we give the hypothesis H3 as followsF1G1 + F2G2 gt 0 where

F1 3σϑ3σminus 10 cos

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 cos

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 cos

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

F2 3σϑ3σminus 10 sin

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 sin

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 sin

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

G1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

G2 α2ϑ2σ0 cos σπ minus α2k3χ

σ0 cos

σπ2

minus α2k31113876 1113877χ0 cos ϑ0ρ0

+ α2ϑ2σ0 sin σπ minus α2k3ϑ

σ0 sin

σπ2

minus α2k31113876 1113877ϑ0 sin ϑ0ρ0

(60)

Lemma 7 Assume that s(ρ) ζ(ρ) + iϖ(ρ) is the root of(42) around ρ ρ0 such that ζ(ρ0) 0 ϖ(ρ0) ϖ0 thenRe[dsdρ]|ρρ0 ϖϖ0 gt 0

Proof of Lemma 7 By (42) we have

dB1(s)

dρ+

dB2(s)

dρe

minus sρminus e

minus sρ ds

dρρ + s1113888 1113889B2(s) 0 (61)

Complexity 9

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

SincedA1(s)

d9 3σs

3σminus 1ds

d9minus 2σ α1 + k3( 1113857s

2σminus 1ds

d9+ σ α1k3 minus k2( 1113857s

σminus 1ds

d9

3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

1113960 1113961ds

d9

dA2(s)

d9 2α1σs

2σminus 1minus α1k3σs

σminus 11113872 1113873

ds

d9

(32)

we obtainds

d91113890 1113891

minus 1

P1(9)

P2(9)minus

9

s (33)

where

P1(s) 3σs3σminus 1

minus 2σ α1 + k3( 1113857s2σminus 1

+ σ α1k3 minus k2( 1113857sσminus 1

+ 2α1σs2σminus 1

minus α1k3σsσminus 1

1113872 1113873eminus s9

P2(s) A2(s)seminus s9

(34)

+us

Reds

d91113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

ReP1(9)

P2(9)1113896 1113897

11138681113868111386811138681113868111386811138681113868990 ψψ0

K1S1 + K2S2

S21 + S2

2

(35)

It follows from H1 that

Reds

d91113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ψψ0

gt 0 (36)

+is proves Lemma 4

Next the hypothesis H2 is given

k3 lt 0

k2k3 gt minus 2k1

k1 lt 0

(37)

Lemma 5 If 9 0 and H2 holds true then system (13) islocally asymptotically stable

Proof of Lemma 5 Under the condition 9 0 (13) becomes

λ3 minus k3λ2

minus k2λ minus k1 0 (38)

In view of H2 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (38) +us Lemma 5 holds true implying theproof

+e following results are established by above discussion

Theorem 1 (under the condition of Lemma 3(ii)) Supposethat H1 and H2 hold true then (i) the equilibrium point of

system (10) at the origin is locally asymptotically stable if0le 9lt 90 and (ii) a Hopf bifurcation will happen near theequilibrium point E1 if 9 90

32 Adding Time-Delayed Feedback Controller to the Sec-ond Equation of Model (2) In this section we add a time-delayed feedback controller α2[w2(t) minus w2(t minus ρ)] tothe second equation of model (2) and then system (2)becomes

dσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)1113858 1113859

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(39)

where α2 is the gain coefficient and ρ is the time delay +elinear equation of equation (39) around E1 (0 0 0) is

dσw1

dtσ w2

dσw2

dtσ α2w2 + w3 minus α2w2(t minus ρ)

dσw3

dtσ k1w1 + k2w2 + k3w3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

+e characteristic equation of (40) takes the followingform

det

sσ minus 1 0

0 sσ minus α2 + α2eminus sρ minus 1

minus k11 minus k2 sρ minus k3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (41)

Hence

B1(s) + B2(s)eminus sρ

0 (42)

where

B1(s) s3σ

minus α2 + k3 + 1( 1113857s2σ

+ α2k3 + k3 minus k2( 1113857sσ

minus k1

B2(s) α2s2σ

minus α2k3sσ

(43)

Assume that s iϑ ϑ(cos(π2) + i sin(π2)) is the rootof (42) then we have

B2R(ϑ)cos ϑρ + B2I(ϑ)sin ζρ minus B1R(ϑ)

B2I(ϑ)cos ϑρ minus B2R(ϕ)sin ϑρ minus B1I(ϑ)1113896 (44)

where

Complexity 7

B1R(ϑ) ϑ3σ cos3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ cos σπ

+ α2k3 + k3 minus k2( 1113857ϑσ cosσπ2

minus k1

B1I(ϑ) ϑ3σ sin3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ sin σπ

+ α2k3 + k3 minus k2( 1113857ϑσ sinσπ2

B2R(ϑ) α2ϑ2ϑ cos ϑπ minus α2k3ϑ

σ cosσπ2

B2I(ϑ) α2ϑ2ϑ sin ϑπ minus α2k3ϑ

σ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

According to (44) we get

cos ϑρ minusB1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)

B22R(ϑ) + B2

2I(ϑ)

sin ϑρ minusB1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)

B22R(ϑ) + B2

2I(ϑ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

Let

s1 cos3σπ2

s2 minus α2 + k3 + 1( 1113857cos σπ

s3 α2k3 + k3 minus k2( 1113857cosσπ2

s4 minus k1

s5 sin3σπ2

s6 minus α2 + k3 + 1( 1113857sin σπ

s7 α2k3 + k3 minus k2( 1113857sinσπ2

s8 α2 cos ϑπ

s9 minus α2k3 cosσπ2

s10 α2 sin ϑπ

s11 minus α2k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

+en

B1R(ϑ) s1ϑ3σ + s2ϑ

2σ + s3ϑσ + s4

B1I(ϑ) s5ϑ3σ + s6ϑ

2σ + s7ϑσ

B2R(ϑ) s8ϑ2σ + s9ϑ

σ

B2I(ϑ) s10ϑ2σ + s11ϑ

σ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(48)

By (46) we have

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

+ B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

B22R(ϑ) + B

22I(ϑ)1113960 1113961

2

(49)

Furthermore

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

t1ϑ5σ

+ t2ϑ4σ

+ t3ϑ3σ

+ t4ϑ2σ

+ t5ϑσ

1113872 11138732

B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

t5ϑ5σ

+ t6ϑ4σ

+ t7ϑ3ϑ

+ t8ϑ2σ

+ t9ϑσ

1113872 11138732

B22R(ϑ) + B

22I(χ)1113960 1113961

2 t10ϑ

4σ+ t11ϑ

3σ+ t12ϑ

2σ+ t13ϑ

σ1113872 1113873

2

(50)

wheret1 s1s8 + s5s10

t2 s2s8 + s1s9 + s6s10 + s5s11

t3 s3s8 + s2s9 + s7s10 + s6s11

t4 s4s8 + s3s9 + s7s11

t5 s4s9

t6 s1s10 minus s5s8

t7 s2s10 + s1s11 minus s5s9 minus s6s8

t8 s3s10 + s2s11 minus s6s9 minus s7s8

t9 s4s10 + s3s11 minus s7s9

t10 s4s11

t11 s28 + s210

t12 2 s8s9 + s10s11( 1113857

t13 s211

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(51)

In view of (49) we have

κ1ϑ10σ

+ κ2ϑ9σ

+ κ3ϑ8σ

+ κ4ϑ7σ

+ κ5ϑ6σ

+ κ6ϑ5σ

+ κ7ϑ4σ

+ κ8ϑ3σ

+ κ9ϑ2σ

0

(52)

where

8 Complexity

κ1 t21 + t26

κ2 2t1t2 + 2t6t7

κ3 t22 + t27 minus t211 + 2t1t3 + 2t6t8

κ4 2 t2t3 + t1t2 + t6t9 + t7t8 minus t11t12( 1113857

κ5 t23 + 2t2t4 + t28 + 2t6t10 + 2t7t9 minus t212 minus 2t11t13

κ6 2t3t4 + 2t8t9

κ7 t24 + t29 + 2t7t10 + 2t8t10 minus t213

κ8 2t9t10 + 2t4t5

κ9 t25 + t210

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(53)

By (52) we have

κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9 0

(54)

Let

K(ϑ) κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9(55)

M(ζ) κ1ζ8

+ κ2ζ7

+ κ3ζ6

+ κ4ζ5

+ κ5ζ4

+ κ6ζ3

+ κ7ζ2

+ κ8ζ1

+ κ9(56)

Lemma 6

(i) Suppose that κj gt 0(j 1 2 9) then the root that haszero real parts does not exist in (42)(ii) Suppose that κ9 lt 0 and existη0 gt 0 which satisfies M(η0)lt 0then (42) possesses at least two pairs of purely imaginary roots

Proof of Lemma 6(i) In view of (55) we havedK(ϑ)

dχ 8κ1σϑ

8σminus 1+ 7κ2σϑ

7σminus 1+ 6κ3σϑ

6σminus 1+ 5κ4σϑ

5σminus 1

+ 4κ5σϑ4σminus 1

+ 3κ6σϑ3σminus 1

+ 2κ7σϑ2σminus 1

+ κ8σϑσminus 1

(57)

In view of κi gt 0(i 1 2 9) we have((dK(ϑ))dϑ) gt 0 forall ϑgt 0 In view of K(0) κ9 gt 0 we knowthat (56) does not possess positive real roots Furthermores 0 is not the root of (42) +is proves Lemma 6(i)

Proof of Lemma 6(ii) In view of M(0) κ9 gt 0 M(η0)lt0(η0 gt 0) and limε⟶+infin((M(ε))dε) +infin exist η01 isin (0 η0)and η02 isin (η0 +infin) such that M(η01) M(η02) 0 Hence(55) has at least two positive real roots and (42) has at leasttwo pairs of purely imaginary roots +is proves Lemma6(ii)

Assume that (54) has eight positive real rootsϑj(j 1 2 8) By (46) we have

ριk 1ϑk

arccos minusB1R ϑm( 1113857B2R ϑk( 1113857 + B1I ϑk( 1113857B2I ϑk( 1113857

B22R ϑk( 1113857 + B2

2I ϑk( 11138571113888 1113889 + 2ιπ1113890 1113891

(58)

where ι 0 1 2 and k 1 2 8 Hence plusmn iϑk is apair of purely imaginary roots of (42) when ρ ριk Let

ρ0 mink128

ρ0k1113966 1113967

ϑ0 ϑ|ρρ0(59)

Next we give the hypothesis H3 as followsF1G1 + F2G2 gt 0 where

F1 3σϑ3σminus 10 cos

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 cos

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 cos

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

F2 3σϑ3σminus 10 sin

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 sin

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 sin

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

G1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

G2 α2ϑ2σ0 cos σπ minus α2k3χ

σ0 cos

σπ2

minus α2k31113876 1113877χ0 cos ϑ0ρ0

+ α2ϑ2σ0 sin σπ minus α2k3ϑ

σ0 sin

σπ2

minus α2k31113876 1113877ϑ0 sin ϑ0ρ0

(60)

Lemma 7 Assume that s(ρ) ζ(ρ) + iϖ(ρ) is the root of(42) around ρ ρ0 such that ζ(ρ0) 0 ϖ(ρ0) ϖ0 thenRe[dsdρ]|ρρ0 ϖϖ0 gt 0

Proof of Lemma 7 By (42) we have

dB1(s)

dρ+

dB2(s)

dρe

minus sρminus e

minus sρ ds

dρρ + s1113888 1113889B2(s) 0 (61)

Complexity 9

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

B1R(ϑ) ϑ3σ cos3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ cos σπ

+ α2k3 + k3 minus k2( 1113857ϑσ cosσπ2

minus k1

B1I(ϑ) ϑ3σ sin3σπ2

minus α2 + k3 + 1( 1113857ϑ2σ sin σπ

+ α2k3 + k3 minus k2( 1113857ϑσ sinσπ2

B2R(ϑ) α2ϑ2ϑ cos ϑπ minus α2k3ϑ

σ cosσπ2

B2I(ϑ) α2ϑ2ϑ sin ϑπ minus α2k3ϑ

σ sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(45)

According to (44) we get

cos ϑρ minusB1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)

B22R(ϑ) + B2

2I(ϑ)

sin ϑρ minusB1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)

B22R(ϑ) + B2

2I(ϑ)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(46)

Let

s1 cos3σπ2

s2 minus α2 + k3 + 1( 1113857cos σπ

s3 α2k3 + k3 minus k2( 1113857cosσπ2

s4 minus k1

s5 sin3σπ2

s6 minus α2 + k3 + 1( 1113857sin σπ

s7 α2k3 + k3 minus k2( 1113857sinσπ2

s8 α2 cos ϑπ

s9 minus α2k3 cosσπ2

s10 α2 sin ϑπ

s11 minus α2k3 sinσπ2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

+en

B1R(ϑ) s1ϑ3σ + s2ϑ

2σ + s3ϑσ + s4

B1I(ϑ) s5ϑ3σ + s6ϑ

2σ + s7ϑσ

B2R(ϑ) s8ϑ2σ + s9ϑ

σ

B2I(ϑ) s10ϑ2σ + s11ϑ

σ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(48)

By (46) we have

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

+ B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

B22R(ϑ) + B

22I(ϑ)1113960 1113961

2

(49)

Furthermore

B1R(ϑ)B2R(ϑ) + B1I(ϑ)B2I(ϑ)1113858 11138592

t1ϑ5σ

+ t2ϑ4σ

+ t3ϑ3σ

+ t4ϑ2σ

+ t5ϑσ

1113872 11138732

B1R(ϑ)B2I(ϑ) minus B1I(ϑ)B2R(ϑ)1113858 11138592

t5ϑ5σ

+ t6ϑ4σ

+ t7ϑ3ϑ

+ t8ϑ2σ

+ t9ϑσ

1113872 11138732

B22R(ϑ) + B

22I(χ)1113960 1113961

2 t10ϑ

4σ+ t11ϑ

3σ+ t12ϑ

2σ+ t13ϑ

σ1113872 1113873

2

(50)

wheret1 s1s8 + s5s10

t2 s2s8 + s1s9 + s6s10 + s5s11

t3 s3s8 + s2s9 + s7s10 + s6s11

t4 s4s8 + s3s9 + s7s11

t5 s4s9

t6 s1s10 minus s5s8

t7 s2s10 + s1s11 minus s5s9 minus s6s8

t8 s3s10 + s2s11 minus s6s9 minus s7s8

t9 s4s10 + s3s11 minus s7s9

t10 s4s11

t11 s28 + s210

t12 2 s8s9 + s10s11( 1113857

t13 s211

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(51)

In view of (49) we have

κ1ϑ10σ

+ κ2ϑ9σ

+ κ3ϑ8σ

+ κ4ϑ7σ

+ κ5ϑ6σ

+ κ6ϑ5σ

+ κ7ϑ4σ

+ κ8ϑ3σ

+ κ9ϑ2σ

0

(52)

where

8 Complexity

κ1 t21 + t26

κ2 2t1t2 + 2t6t7

κ3 t22 + t27 minus t211 + 2t1t3 + 2t6t8

κ4 2 t2t3 + t1t2 + t6t9 + t7t8 minus t11t12( 1113857

κ5 t23 + 2t2t4 + t28 + 2t6t10 + 2t7t9 minus t212 minus 2t11t13

κ6 2t3t4 + 2t8t9

κ7 t24 + t29 + 2t7t10 + 2t8t10 minus t213

κ8 2t9t10 + 2t4t5

κ9 t25 + t210

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(53)

By (52) we have

κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9 0

(54)

Let

K(ϑ) κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9(55)

M(ζ) κ1ζ8

+ κ2ζ7

+ κ3ζ6

+ κ4ζ5

+ κ5ζ4

+ κ6ζ3

+ κ7ζ2

+ κ8ζ1

+ κ9(56)

Lemma 6

(i) Suppose that κj gt 0(j 1 2 9) then the root that haszero real parts does not exist in (42)(ii) Suppose that κ9 lt 0 and existη0 gt 0 which satisfies M(η0)lt 0then (42) possesses at least two pairs of purely imaginary roots

Proof of Lemma 6(i) In view of (55) we havedK(ϑ)

dχ 8κ1σϑ

8σminus 1+ 7κ2σϑ

7σminus 1+ 6κ3σϑ

6σminus 1+ 5κ4σϑ

5σminus 1

+ 4κ5σϑ4σminus 1

+ 3κ6σϑ3σminus 1

+ 2κ7σϑ2σminus 1

+ κ8σϑσminus 1

(57)

In view of κi gt 0(i 1 2 9) we have((dK(ϑ))dϑ) gt 0 forall ϑgt 0 In view of K(0) κ9 gt 0 we knowthat (56) does not possess positive real roots Furthermores 0 is not the root of (42) +is proves Lemma 6(i)

Proof of Lemma 6(ii) In view of M(0) κ9 gt 0 M(η0)lt0(η0 gt 0) and limε⟶+infin((M(ε))dε) +infin exist η01 isin (0 η0)and η02 isin (η0 +infin) such that M(η01) M(η02) 0 Hence(55) has at least two positive real roots and (42) has at leasttwo pairs of purely imaginary roots +is proves Lemma6(ii)

Assume that (54) has eight positive real rootsϑj(j 1 2 8) By (46) we have

ριk 1ϑk

arccos minusB1R ϑm( 1113857B2R ϑk( 1113857 + B1I ϑk( 1113857B2I ϑk( 1113857

B22R ϑk( 1113857 + B2

2I ϑk( 11138571113888 1113889 + 2ιπ1113890 1113891

(58)

where ι 0 1 2 and k 1 2 8 Hence plusmn iϑk is apair of purely imaginary roots of (42) when ρ ριk Let

ρ0 mink128

ρ0k1113966 1113967

ϑ0 ϑ|ρρ0(59)

Next we give the hypothesis H3 as followsF1G1 + F2G2 gt 0 where

F1 3σϑ3σminus 10 cos

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 cos

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 cos

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

F2 3σϑ3σminus 10 sin

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 sin

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 sin

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

G1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

G2 α2ϑ2σ0 cos σπ minus α2k3χ

σ0 cos

σπ2

minus α2k31113876 1113877χ0 cos ϑ0ρ0

+ α2ϑ2σ0 sin σπ minus α2k3ϑ

σ0 sin

σπ2

minus α2k31113876 1113877ϑ0 sin ϑ0ρ0

(60)

Lemma 7 Assume that s(ρ) ζ(ρ) + iϖ(ρ) is the root of(42) around ρ ρ0 such that ζ(ρ0) 0 ϖ(ρ0) ϖ0 thenRe[dsdρ]|ρρ0 ϖϖ0 gt 0

Proof of Lemma 7 By (42) we have

dB1(s)

dρ+

dB2(s)

dρe

minus sρminus e

minus sρ ds

dρρ + s1113888 1113889B2(s) 0 (61)

Complexity 9

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

κ1 t21 + t26

κ2 2t1t2 + 2t6t7

κ3 t22 + t27 minus t211 + 2t1t3 + 2t6t8

κ4 2 t2t3 + t1t2 + t6t9 + t7t8 minus t11t12( 1113857

κ5 t23 + 2t2t4 + t28 + 2t6t10 + 2t7t9 minus t212 minus 2t11t13

κ6 2t3t4 + 2t8t9

κ7 t24 + t29 + 2t7t10 + 2t8t10 minus t213

κ8 2t9t10 + 2t4t5

κ9 t25 + t210

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(53)

By (52) we have

κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9 0

(54)

Let

K(ϑ) κ1ϑ8σ

+ κ2ϑ7σ

+ κ3ϑ6σ

+ κ4ϑ5σ

+ κ5ϑ4σ

+ κ6ϑ3σ

+ κ7ϑ2σ

+ κ8ϑσ

+ κ9(55)

M(ζ) κ1ζ8

+ κ2ζ7

+ κ3ζ6

+ κ4ζ5

+ κ5ζ4

+ κ6ζ3

+ κ7ζ2

+ κ8ζ1

+ κ9(56)

Lemma 6

(i) Suppose that κj gt 0(j 1 2 9) then the root that haszero real parts does not exist in (42)(ii) Suppose that κ9 lt 0 and existη0 gt 0 which satisfies M(η0)lt 0then (42) possesses at least two pairs of purely imaginary roots

Proof of Lemma 6(i) In view of (55) we havedK(ϑ)

dχ 8κ1σϑ

8σminus 1+ 7κ2σϑ

7σminus 1+ 6κ3σϑ

6σminus 1+ 5κ4σϑ

5σminus 1

+ 4κ5σϑ4σminus 1

+ 3κ6σϑ3σminus 1

+ 2κ7σϑ2σminus 1

+ κ8σϑσminus 1

(57)

In view of κi gt 0(i 1 2 9) we have((dK(ϑ))dϑ) gt 0 forall ϑgt 0 In view of K(0) κ9 gt 0 we knowthat (56) does not possess positive real roots Furthermores 0 is not the root of (42) +is proves Lemma 6(i)

Proof of Lemma 6(ii) In view of M(0) κ9 gt 0 M(η0)lt0(η0 gt 0) and limε⟶+infin((M(ε))dε) +infin exist η01 isin (0 η0)and η02 isin (η0 +infin) such that M(η01) M(η02) 0 Hence(55) has at least two positive real roots and (42) has at leasttwo pairs of purely imaginary roots +is proves Lemma6(ii)

Assume that (54) has eight positive real rootsϑj(j 1 2 8) By (46) we have

ριk 1ϑk

arccos minusB1R ϑm( 1113857B2R ϑk( 1113857 + B1I ϑk( 1113857B2I ϑk( 1113857

B22R ϑk( 1113857 + B2

2I ϑk( 11138571113888 1113889 + 2ιπ1113890 1113891

(58)

where ι 0 1 2 and k 1 2 8 Hence plusmn iϑk is apair of purely imaginary roots of (42) when ρ ριk Let

ρ0 mink128

ρ0k1113966 1113967

ϑ0 ϑ|ρρ0(59)

Next we give the hypothesis H3 as followsF1G1 + F2G2 gt 0 where

F1 3σϑ3σminus 10 cos

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 cos

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 cos

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

F2 3σϑ3σminus 10 sin

(3σ minus 1)π2

minus 2σ α2 + k3 + 1( 1113857ϑ2σminus 10 sin

(2σ minus 1)π2

+ σ α2k3 + k3 minus k2( 1113857ϑσminus 10 sin

(σ minus 1)π2

+ 2σα2ϑ2σminus 10 cos

(2σ minus 1)π2

minus α2k3ϑσminus 10 cos

(σ minus 1)π2

1113890 1113891sin ϑ0ρ0

+ 2σα2ϑ2σminus 10 sin

(2σ minus 1)π2

minus minus α2k3ϑσminus 10 sin

(σ minus 1)π2

1113890 1113891cos ϑ0ρ0

G1 α1χ2σ0 cos σπ minus α1k3χ

σ0 cos

σπ2

minus α1k21113876 1113877χ0 sin χ090

+ α1χ2σ0 sin σπ minus α1k3χ

σ0 sin

σπ2

minus α1k21113876 1113877χ0 cos χ090

G2 α2ϑ2σ0 cos σπ minus α2k3χ

σ0 cos

σπ2

minus α2k31113876 1113877χ0 cos ϑ0ρ0

+ α2ϑ2σ0 sin σπ minus α2k3ϑ

σ0 sin

σπ2

minus α2k31113876 1113877ϑ0 sin ϑ0ρ0

(60)

Lemma 7 Assume that s(ρ) ζ(ρ) + iϖ(ρ) is the root of(42) around ρ ρ0 such that ζ(ρ0) 0 ϖ(ρ0) ϖ0 thenRe[dsdρ]|ρρ0 ϖϖ0 gt 0

Proof of Lemma 7 By (42) we have

dB1(s)

dρ+

dB2(s)

dρe

minus sρminus e

minus sρ ds

dρρ + s1113888 1113889B2(s) 0 (61)

Complexity 9

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

SincedB1(s)

dρ 3σs

3σminus 1ds

d9minus 2σ α2 + k3 + 1( 1113857s

2σminus 1ds

dρ

+ σ α2k3 + k3 minus k2( 1113857sσminus 1ds

dρ

11138903σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ + σ α2k3 + k3 minus k2( 1113857sσminus 1

1113891ds

dρ

dB2(s)

dρ 2α2σs

2σminus 1minus α2k3σs

σminus 11113872 1113873

ds

dρ

(62)

we have

ds

dρ1113890 1113891

minus 1

Q1(ρ)

Q2(ρ)minusρs (63)

where

Q1(s) 3σs3σminus 1

minus 2σ α2 + k3 + 1( 1113857s2σminus 1

+ σ α2k3 + k3 minus k2( 1113857sσminus 1

+ 2α2σs2σminus 1

minus α2k3σsσminus 1

1113872 1113873eminus s9

Q2(s) B2(s)seminus sρ

(64)

+us

Reds

dρ1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

ReQ1(ρ)

Q2(ρ)1113896 1113897

11138681113868111386811138681113868111386811138681113868ρρ0ϖϖ0

F1G1 + F2G2

G21 + G2

2

(65)

By H3 we have

Reds

dρ1113890 1113891

minus 1⎧⎨

⎩

⎫⎬

⎭

111386811138681113868111386811138681113868111386811138681113868ρρ0 ϖϖ0

gt 0 (66)

+is proves Lemma 7

Next we give the hypothesis H4 as follows

k3 + 1lt 0

k2 + 1( 1113857 k3 minus k2( 1113857lt minus 2k1

k1 lt 0

(67)

Lemma 8 If ρ 0 and H4 holds true then system (42) islocally asymptotically stable

Proof of Lemma 8 If ρ 0 then (42) takes the following form

λ3 minus k3λ2

minus k2λ minus k1 0 (68)

In view of H4 |arg(λi)|gt ((σπ)2)(i 1 2 3) where λi

is the root of (68) +erefore Lemma 8 holds true implyingthe proof Theorem 2 (in addition to the condition of Lemma6(ii)) Assume that H3 and H4 are fulfilled then (i) theequilibrium point of model (39) at the origin is locally

asymptotically stable if 0le ρlt ρ0 and (ii) a Hopf bifurcationwill happen near the equilibrium point E1 if ρ ρ0

Remark 2 Although there are many papers [17 18 21 23]that deal with the chaos control by applying the time-delayedfeedback controller they only focus on the integer-orderdifferential systems In this paper we handle the chaoscontrol for the fractional-order differential model All thederived results and analysis ways for the chaos control ofinteger-order differential systems cannot be transferred to(2) to control the chaotic phenomenon From this viewpointwe can say that the results of this paper are completelyinnovative and supplement the earlier investigation

Remark 3 By computer simulations and maximum Lyapu-nov exponent we show that model (2) has chaotic behavior Ifthe time delay is introduced into the controlled chaotic model(10) or (39) then we can control the chaotic phenomenon ofthe origin system (2) under some suitable conditions Also wecan examine the chaotic nature of the fractional-order systemby maximum Lyapunov exponent and computer simulations

Remark 4 Adding the time-delayed feedback controller tothe first equation and the second equation of model (2) isonly a try Of course we can add the controller to the thirdequation +rough the later analysis we can check whetherthis control technique is right or not

Remark 5 Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesimodel have the same form the dynamical behavior and theresearch method on stability and Hopf bifurcation are verydifferent because of introduction and the variation offractional order According to some previous related papers(see [17ndash19 21 38 39]) we can see that the dynamicalbehavior and the research method on stability and Hopfbifurcation for fractional-order differential systems are verydifferent from those for integer-order differential systems

4 Two Examples

In this section we use a new predictor-corrector method(NPCM) [33] to carry out computer simulations

Example 1 +e following model is givend087w1

dt087 w2 + α1 w1(t) minus w1(t minus 9)1113858 1113859

d087w2

dt087 w3

d087w3

dt087 minus 6w1 minus 292w2 minus 12w3 + w21

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(69)

Obviously model (69) possesses the zero equilibriumpoint Let α1 22 the critical frequency χ0 12355 andthe bifurcation point 90 208 We can easily check that the

10 Complexity

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

hypotheses of eorem 1 are fullled Figure 3 indicates thatwhen 9 isin [0 208) the zero equilibrium point of model (69)is locally asymptotically stable Figure 4 shows that when9 isin [208+infin) a Hopf bifurcation happens for model (69)Both cases illustrate that the chaotic behavior can be con-trolled by applying the designed time-delayed controller22[w2(t) minus w2(t minus 9)] e bifurcation diagram is presentedin Figure 5

Example 2 e following model is givendσw1

dtσ w2

dσw2

dtσ w3 + α2 w2(t) minus w2(t minus ρ)[ ]

dσw3

dtσ k1w1 + k2w2 + k3w3 + w

21

(70)

It is easy to see that model (70) possesses the zeroequilibrium point Let α2 22 the critical frequency

ϑ0 23511 and the bifurcation point ρ0 077 We caneasily check that the hypotheses of eorem 1 are satisedFigure 6 indicates that when ρ isin [0 077) the zero equi-librium point of model (70) is locally asymptoticallystable Figure 7 shows that when ρ isin [077+infin) a Hopfbifurcation happens for model (70) Both cases manifestthat the chaotic behavior of model (70) can be suppressedby using the time-delayed feedback controller22[w2(t) minus w2(t minus ρ)]

Remark 6 According to the computer simulation results ofmodel (69) and model (70) we can see that when the timedelay remains in a suitable range the controller chaoticmodel is locally asymptotically stable and when the timedelay crosses a certain critical value then a Hopf bi-furcation appears near the equilibrium point of thecontrolled model Based on the simulation results abovewe can conclude that the time-delayed feedback controllerin model (70) is more eective for the chaos control thanthat in model (69) since the delay in model (70) is less thanthe delay in model (69)

minus02

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120minus25

minus2

minus15

minus1

minus05

0

05

1

15

t

w 2 (t

)

(b)

minus4

minus3

minus2

minus1

0

1

2

3

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus050

051

15

minus4minus2

02

minus4

minus2

0

2

4

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 3 Computer simulations of model (69) when 9 19lt 90 208 and the initial value is (1 1 1)

Complexity 11

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

5 Conclusions

In real life chaos control has become a hot issue During thepast decades a lot of control techniques are proposed tosuppress the chaotic phenomenon But all the controlstrategies are only concerned with the integer-order dif-ferential chaotic systems In this paper we adopt two time-delayed feedback controllers to control the same fractional-

order chaotic Genesio-Tesi model e research results showthat both time-delayed feedback controllers can eectivelycontrol the chaotic behavior of the fractional-order chaoticGenesio-Tesi model By analyzing the characteristic equa-tion of the controlled fractional-order chaotic Genesio-Tesimodel we establish two sucient conditions to ensure thestability and the appearance of Hopf bifurcation of theinvolved chaotic Genesio-Tesi model In addition the

minus3

minus2

minus1

0

1

2

3

4

0 20 40 60 80 100 120t

w 1 (t

)

(a)

0 20 40 60 80 100 120t

minus4

minus3

minus2

minus1

0

1

2

3

w 3 (t

)

(b)

minus8

minus6

minus4

minus2

0

2

4

6

8

0 20 40 60 80 100 120t

w 3 (t

)

(c)

minus4 minus20

24

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t) w1 (t)

(d)

Figure 4 Computer simulations of model (69) when 9 23gt 90 208 and the initial value is (1 1 1)

0 1 2 3 4 5 6 7minus3

minus2

minus1

0

1

2

3

Time delay

w 1

Figure 5 Bifurcation diagram with respect to the time delay 9 for model (69) e horizontal axis is 9 and the vertical axis is w1(t)

12 Complexity

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

0 50 100 150 200minus15

minus1

minus05

0

05

1

15

2

t

w 1 (t

)

(a)

0 50 100 150 200t

minus2

minus15

minus1

minus05

0

05

1

15

w 2 (t

)

(b)

0 50 100 150 200t

minus5minus4minus3minus2minus1

012345

w 3 (t

)

(c)

minus2minus1 0

12

minus2minus1

01

2minus5

0

5

w 3 (t

)

w2 (t) w2 (t)

(d)

Figure 6 Computer simulations of model (70) when ρ 070lt ρ0 077 and the initial value is (1 1 1)

0 50 100 150 200t

minus2minus15

minus1minus05

005

115

252

w 1 (t

)

(a)

0 50 100 150 200t

minus3

minus2

minus1

0

1

2

3

w 2 (t

)

(b)

0 50 100 150 200t

minus8minus6minus4minus2

02468

10

w 3 (t

)

(c)

minus2 minus1 0 1 2 3

minus4minus2

02

4minus10

minus5

0

5

10

w 3 (t

)

w2 (t)

w1 (t)

(d)

Figure 7 Computer simulations of model (70) when ρ 082gt ρ0 077 and the initial value is (1 1 1)

Complexity 13

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

control technique can be widely applied in numerousfractional-order chaotic models

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (No 61673008) Project of High-LevelInnovative Talents of Guizhou Province ([2016]5651) MajorResearch Project of +e Innovation Group of +e EducationDepartment of Guizhou Province ([2017]039) InnovativeExploration Project of Guizhou University of Finance andEconomics ([2017]5736-015) Project of Key Laboratory ofGuizhou Province with Financial and Physical Features([2017]004) Hunan Provincial Key Laboratory of Mathe-matical Modeling and Analysis in Engineering (ChangshaUniversity of Science amp Technology) (2018MMAEZD21)University Science and Technology Top Talents Project ofGuizhou Province (KY[2018]047) and Guizhou Universityof Finance and Economics (2018XZD01)

References

[1] J H Park and O M Kwon ldquoLMI optimization approach tostabilization of time-delay chaotic systemsrdquo Chaos Solitons ampFractals vol 23 no 2 pp 445ndash450 2005

[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquoPhysical Review Letters vol 64 no 11 pp 1196ndash1199 1990

[3] C-C Wang and J-P Su ldquoA new adaptive variable structurecontrol for chaotic synchronization and secure communi-cationrdquo Chaos Solitons amp Fractals vol 20 no 5 pp 967ndash9772004

[4] X-S Yang and G Chen ldquoSome observer-based criteria fordiscrete-time generalized chaos synchronizationrdquo ChaosSolitons amp Fractals vol 13 no 6 pp 1303ndash1308 2002

[5] X Wu and J Lu ldquoParameter identification and backsteppingcontrol of uncertain Lu systemrdquo Chaos Solitons amp Fractalsvol 18 no 4 pp 721ndash729 2003

[6] E-W Bai and K E Lonngren ldquoSequential synchronization oftwo lorenz systems using active controlrdquo Chaos Solitons ampFractals vol 11 no 7 pp 1041ndash1044 2000

[7] U E Kocamaz B Cevher and Y Uyaroglu ldquoControl andsynchronization of chaos with sliding mode control based oncubic reaching rulerdquo Chaos Solitons amp Fractals vol 105pp 92ndash98 2017

[8] G Chen ldquoA simple adaptive feedback control method forchaos and hyper-chaos controlrdquo Applied Mathematics andComputation vol 217 no 17 pp 7258ndash7264 2011

[9] Q Din ldquoComplexity and chaos control in a discrete-timeprey-predator modelrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 49 pp 113ndash134 2017

[10] A Singh and S Gakkhar ldquoControlling chaos in a food chainmodelrdquo Mathematics and Computers in Simulation vol 115pp 24ndash36 2015

[11] J-J Yan C-Y Chen and J S-H Tsai ldquoHybrid chaos controlof continuous unified chaotic systems using discrete ripplingsliding mode controlrdquo Nonlinear Analysis Hybrid Systemsvol 22 pp 276ndash283 2016

[12] J Yang and L Zhao ldquoBifurcation analysis and chaos control ofthe modified Chuarsquos circuit systemrdquo Chaos Solitons ampFractals vol 77 pp 332ndash339 2015

[13] A K O Tiba A F R Araujo and F R Aluizio ldquoControlstrategies for Hopf bifurcation in a chaotic associativememoryrdquo Neurocomputing vol 323 pp 157ndash174 2019

[14] V K Yadav V K Shukla and S Das ldquoDifference syn-chronization among three chaotic systems with exponentialterm and its chaos controlrdquo Chaos Solitons amp Fractalsvol 124 pp 36ndash51 2019

[15] X Lian J Liu J Zhang and C Wang ldquoChaotic motion andcontrol of a tethered-sailcraft system orbiting an asteroidrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 77 pp 203ndash224 2019

[16] K +orsen T Drengstig and P Ruoff ldquo+e effect of integralcontrol in oscillatory and chaotic reaction kinetic networksrdquoPhysica D Nonlinear Phenomena vol 393 pp 38ndash46 Inpress 2019

[17] C Xu and Q Zhang ldquoOn the chaos control of the qi systemrdquoJournal of Engineering Mathematics vol 90 no 1 pp 67ndash812015

[18] C Xu and Y Wu ldquoBifurcation and control of chaos in achemical systemrdquo Applied Mathematical Modelling vol 39no 8 pp 2295ndash2310 2015

[19] C Xu M Liao and P Li ldquoBifurcation control for a fractional-order competition model of internet with delaysrdquo NonlinearDynamics vol 95 no 4 pp 3335ndash3356 2019

[20] R Genesio and A Tesi ldquoHarmonic balance methods for theanalysis of chaotic dynamics in nonlinear systemsrdquo Auto-matica vol 28 no 3 pp 531ndash548 1992

[21] J Guan F Y Chen and G XWang ldquoBifurcation analysis andchaos control in Genesio systemwith delayed feedbackrdquo ISRNMathematical Physics vol 2012 Article ID 843962 12 pages2012

[22] Y-J Sun ldquoChaos synchronization of uncertain genesio-tesichaotic systems with dead zone nonlinearityrdquo Physics LettersA vol 373 no 36 pp 3273ndash3276 2009

[23] J H Park ldquoExponential synchronization of the genesio-tesichaotic system via a novel feedback controlrdquo Physica Scriptavol 76 no 6 pp 617ndash622 2007

[24] J H Park ldquoFurther results on functional projective syn-chronization of genesio-tesi chaotic systemrdquo Modern PhysicsLetters B vol 23 no 15 pp 1889ndash1895 2009

[25] L Zhou and F Chen ldquoHopf bifurcation and Sirsquolnikov chaos ofgenesio systemrdquo Chaos Solitons amp Fractals vol 40 no 3pp 1413ndash1422 2009

[26] Z Wang X Huang and G Shi ldquoAnalysis of nonlinear dy-namics and choas in a fractional order financial sytem withtime delayrdquo Computers amp Mathematics with Applicationsvol 62 no 2 pp 1531ndash1539 2011

[27] X B Gui C Tong and L Y Qin ldquoComplexity evolvement ofa chaotic fractional-order financial systemrdquo Acta PhysicaSinica vol 60 article 048901 no 4 p 6 2011

[28] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNon-trivial solutions for a fractional advection dispersion equationin anomalous diffusionrdquo Applied Mathematics Letters vol 66pp 1ndash8 2017

[29] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

14 Complexity

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

[30] B Bandyopadhyay and S Kamal Stabliization and Control ofFractional Order Systems A Sliding Mode Approach Vol 317Springer Heidelberg Germany 2015

[31] D Matignon ldquoStability results for fractional differentialequations with applications to control processing compu-tational engineering in systems and application multi-con-ference IMACSrdquo in Proceedings of the IEEE-SMCProceedings vol 2 pp 963ndash968 Lille France July 1996

[32] W Deng C Li and J Lu ldquoStability analysis of linear frac-tional differential system with multiple time delaysrdquo Non-linear Dynamics vol 48 no 4 pp 409ndash416 2007

[33] V Daftardar-Gejji Y Sukale and S Bhalekar ldquoA new pre-dictor-corrector method for fractional differential equationsrdquoApplied Mathematics and Computation vol 244 pp 158ndash1822014

[34] Y Fan X Huang H Shen and J Cao ldquoSwitching event-triggered control for global stabilization of delayed mem-ristive neural networks an exponential attenuation schemerdquoNeural Networks vol 117 pp 216ndash224 2019

[35] X Wang Z Wang and H Shen ldquoDynamical analysis of adiscrete-time SIS epidemic model on complex networksrdquoApplied Mathematics Letters vol 94 pp 292ndash299 2019

[36] Y Fan X Huang Z Wang and Y Li ldquoNonlinear dynamicsand chaos in a simplified memristor-based fractional-orderneural network with discontinuous memductance functionrdquoNonlinear Dynamics vol 93 no 2 pp 611ndash627 2018

[37] Z Wang X Wang Y Li and X Huang ldquoStability and Hopfbifurcation of fractional-order complex-valued single neuronmodel with time delayrdquo International Journal of Bifurcationand Chaos vol 27 no 13 article 1750209 2017

[38] L Li Z Wang Y Li H Shen and J Lu ldquoHopf bifurcationanalysis of a complex-valued neural network model withdiscrete and distributed delaysrdquo Applied Mathematics andComputation vol 330 pp 152ndash169 2018

[39] Z Wang L Li Y Li and Z Cheng ldquoStability and hopf bi-furcation of a three-neuron network with multiple discreteand distributed delaysrdquo Neural Processing Letters vol 48no 3 pp 1481ndash1502 2018

[40] Y Fan X Huang Z Wang and Y Li ldquoImproved quasi-synchronization criteria for delayed fractional-order mem-ristor-based neural networks via linear feedback controlrdquoNeurocomputing vol 306 pp 68ndash79 2018

[41] S Chen and X Tang ldquoImproved results for Klein-Gordon-Maxwell systems with general nonlinearityrdquo Discrete ampContinuous Dynamical Systems-A vol 38 no 5 pp 2333ndash2348 2018

[42] S Chen and X Tang ldquoGeometrically distinct solutions forKlein-Gordon-Maxwell systems with super-linear non-linearitiesrdquo Applied Mathematics Letters vol 90 pp 188ndash1932019

[43] X H Tang and S T Chen ldquoGround state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with generalpotentialsrdquo Calculus of Variations and Partial DifferentialEquations vol 56 no 4 pp 1ndash25 2017

[44] X H Tang and S T Chen ldquoGround state solutions ofSchrodinger-Poisson systems with variable potential andconvolution nonlinearityrdquo Journal of Mathematical Analysisand Applications vol 473 no 1 pp 87ndash111 2019

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom