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Research ArticleAnalysis Stabilization and DSP-Based Implementation of aChaotic System with Nonhyperbolic Equilibrium

Xuan-Bing Yang12 Yi-Gang He 13 Chun-Lai Li 2 and Chang-Qing Liu1

1School of Electrical Engineering and Automation Hefei University of Technology Hefei 230009 China2School of Information Science and Engineering Hunan Institute of Science and Technology Yueyang 414006 China3School of Electrical Engineering and Automation Wuhan University Wuhan 430072 China

Correspondence should be addressed to Yi-Gang He 18655136887163com and Chun-Lai Li hnistlichl163com

Received 31 August 2019 Revised 21 October 2019 Accepted 13 November 2019 Published 8 January 2020

Guest Editor Viet-anh Pham

Copyright copy 2020 Xuan-Bing Yang et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

is paper reports an autonomous dynamical system and it nds that one nonhyperbolic zero equilibrium and two hyperbolicnonzero equilibria coexist in this system us it is dicult to demonstrate the existence of chaos by Silrsquonikov theoremConsequently the topological horseshoe theory is adopted to rigorously prove the chaotic behaviors of the system in the phasespace of Poincare map en a single control scheme is designed to stabilize the dynamical system to its zero-equilibrium pointBesides to verify the theoretical analyses physically the attractor and stabilization scheme are further realized via DSP-based technique

1 Introduction

Chaotic behaviors exist widely in biology engineeringeconomy and many other scientic disciplines [1ndash4]Chaotic systems have the properties of unpredictabilitytopological mixing ergodicity and sensitivity to their initialvalues and control parameters Owing to the noise-likespectrum and broad-band chaotic signals are potentiallyapplicable in engineering such as random sequence gener-ation [5] secure communication [6] image encryption[7 8] signal detection [9] radar and sonar systems [10] andso on erefore it is signicant to design and analyze newchaotic systems In 1963 Lorenz proposed the rst three-dimensional chaotic system when studying earthrsquos atmo-spheric convection [11] en many low-dimensionalmathematical models with positive Lyapunov exponentshave been introduced along with the analysis of a rich classof dynamical behaviors New examples continue to be re-ported for publication in nonlinear dynamics journals forthe physical applications Nevertheless it is necessary todevelop chaotic systems with simple structure and richdynamical behaviors from the perspective of application

As a striking chaos theory with symbolic dynamicsobtained by Kennedy in continuous map [12] the topo-logical horseshoe can provide an impactful tool for provingchaotic dynamics in hyperbolic or nonhyperbolic systemUp to present many noteworthy theoretical progresses havebeen extended in nding the existence of horseshoe Forexample Yang introduced the remarkable criteria fornding the topological horseshoe in noncontinuous map[13 14] which has been successfully applied to somepractical systems for verifying chaos [15 16] Li presented anew method with three steps for nding horseshoes indynamical systems by using several simple results on to-pological horseshoes [17] However it is still a tough work tond a topological horseshoe in a practical chaotic system[18]

It is impossible to chronically predict the future behaviorof chaotic system but one can stabilize the future behaviorinto a certain range by using control technologye seminalattempt of controlling chaos is the well-known OGYmethod which applies small perturbations to system pa-rameter to keep the system close to the target periodic orbit[19] However the experimental implementation of this

HindawiComplexityVolume 2020 Article ID 6786832 11 pageshttpsdoiorg10115520206786832

method is restricted by the level of noise in the experimentaldata for the sake of the discrete nature of the control signalen a continuous delayed feedback scheme is proposedconsequently [20] e control signal is the perturbations ofthe difference between the states of the current system andone period of the target orbit in the past erefore theintensity of the perturbations will vanish when the systemevolves to the desired orbit Since then a wide variety ofcontrol approaches emerged for different applications suchas backstepping control [21] sliding mode control [22]sampled-data control [23] and multiswitching combinationcontrol [24] just to name a few However the presentschemes focused on the control problem need severalcontrollers From the points of both practical applicationand theoretical research it is significant to design simple yetexecutable control technique

In this paper an autonomous chaotic system with asimple algebraic structure of six terms is proposed Basicdynamical properties of the system including equilibriumpoint phase portrait Poincare map parameter bifurcationand Lyapunov exponent are studied in theory and nu-merical simulation It is found that this system exhibitsfruitful dynamic behaviors of dense periodic windows andcoexistence of nonhyperbolic and hyperbolic equilibriumpoints And to rigorously verify the emergence of chaos ofthis system in theory the topological horseshoe is investi-gated in the phase space of Poincare mapen based on theLyapunov stability criterion a single control scheme isdesigned to stabilize the chaotic system to its zero-equi-librium point e implementation scheme of attractors andcontrol scheme are discussed in detail and realized via DSP-based technique confirming the validity and enforceabilityof the theoretical scheme

2 The Proposed Dynamical System

e autonomous dynamical system considered here is givenby

_x1 minus ax1 + bx2

_x2 ex2 minus fx1x3

_x3 minus cx33 + dx1x2

⎧⎪⎪⎨

⎪⎪⎩(1)

Differential equations of system (1) are simple with acubic term and two quadratic cross-product terms esystem parameters a b c d e and f are all positiveconstants

21DissipativityandExistenceofAttractor We first considerthe general condition of dissipativity to ensure the chaoticproperty

nablaV z _x1

x1+

z _x2

x2+

z _x3

x3 minus a + e minus 3cx

23 (2)

erefore system (1) would be dissipative and willconverge to a subset of measure zero volume according todVdt e(minus a+eminus 3cx2

3) when satisfying minus a + e minus 3cx23 lt 0 is

means that the volume will become V(0)e(minus a+eminus 3cx23)t at time

t through the flow generated by the system for an initialvolume V(0) erefore there exists an attractor in system(1) with minus a + e minus 3cx2

3 lt 0

22 Equilibrium Points and Stability Considering the con-dition of equilibrium point _x1 0 _x2 0 and _x3 0 weobtain three equilibrium points of system (1) as follows

P0(0 0 0)

P1a

b

ce3

df3

1113971

a2

b2

ce3

df3

1113971

ae

bf⎛⎝ ⎞⎠

P2 minusa

b

ce3

df3

1113971

minusa2

b2

ce3

df3

1113971

ae

bf⎛⎝ ⎞⎠

(3)

When selecting a 1 b 6 c 5 d 1 e 3 and f 2 theequilibrium points and the corresponding eigenvalues areshown in Table 1 including the type of equilibrium points

From Table 1 it is known that the three equilibriumpoints are all unstable with stable manifold and unstablemanifold According to [25] since the characteristic value λ3of equilibrium point P0 equals to zero the equilibrium pointis nonhyperbolic type However equilibrium points P1 andP2 are hyperbolic since all the real parts of the correspondingeigenvalues of these two equilibrium points are nonzeroerefore this is a chaotic system in which hyperbolic andnonhyperbolic equilibrium points coexist And this kind ofchaotic system does not belong to Silrsquonikov sense of thechaotic system and it is difficult to prove the existence ofchaos by Silrsquonikov theorem At present chaotic systems withspecial features such as chaotic systems with nonequilib-rium with multistability and with hyperbolic and non-hyperbolic equilibrium coexisting have attracted extensiveattention by researchers [26 27]

23 Phase Portrait and Chaotic Properties When selectinga 1 b 6 c 5 d 1 e 3 f 2 and initial values x(0) (001 001 005) the Lyapunov exponents of system (1)are depicted in Figures 1(a) and 1(b) with the values as0123017 0000029 and minus 11414797 According to the threeLyapunov exponents the KaplanndashYorke dimension isDKY 2 + (0123017 + 0000029)11414797 20108 ere-fore the KaplanndashYorke dimension is fractional e cor-responding chaotic phase diagrams further reveal that theproposed system displays complicated chaotic behaviors asdepicted in Figure 2

As an important analytical technique Poincare map canreflect the bifurcation and folding properties of chaosSelecting the parameter values a 1 b 6 c 5 d 1 e 3and f 2 and taking the crossing planes x3 0 x1 0 andx2 0 we obtain the corresponding Poincare maps illus-trated in Figure 3 We can see the attractor structure fromthe dense dots

24 Influence of Parameter Variation It is found that system(1) exhibits complicated dynamical behaviors with the

2 Complexity

variation of system parameters As explication we onlyconsider the variation of parameters c and f in thissection

When fixing parameters a 1 b 6 d 1 e 3 f 2and letting c vary from 2 to 8 the corresponding bifur-cation diagram of state variable x3 and the maximumLyapunov exponent of system (1) versus c of numericalcalculation are shown in Figures 4(a) and 4(b) respec-tively As we know that system (1) shows rich dynamicalbehaviors ranging from stable equilibrium points peri-odic orbits to chaotic oscillations depending on the pa-rameter values Furthermore there emerge many visibleperiodic windows in the chaotic region en we fix pa-rameters a 1 b 6 c 5 d 1 and e 3 while letting fvary from 1 to 3 Figures 5(a) and 5(b) depict the bifur-cation diagram of state variable x3 and the maximumLyapunov exponent of system (1) versus parameter f re-spectively It is known that with the variation of parameterf system (1) ranges from stable equilibrium points pe-riodic orbits to chaotic oscillations and there emergesmany densely distributed periodic windows in the chaoticregion also showing the rich dynamics e variation ofthe properties of system (1) with system parameters c and fis of great importance in image encryption

3 Topological Horseshoe in theDynamical System

31 Review of Topological Horseshoe 2eorems It is still achallenge to find topological horseshoes in a concrete sys-tem especially to select a suitable quadrilateral in the crosssection Before studying the horseshoe embedded in thedynamical system some theorems on topological horseshoeare reviewed below [12ndash14 17]

LetD be a compact subset of S which is ametric space andthere exists m mutually disjoint compact subsets D1 D2 Dm of D

Definition 1 Let D1i D2

i sub Di be two fixed disjoint compactsubsets with 1le ilem If ccapD1

i and ccapD2i are compact and

nonempty we say that the connected subset c of Di connectsD1

i and D2i and denote this by D1

iharrc

D2i

Definition 2 Let c be a connected subset of Di we say thatf(c) is suitably across Di with respect to D1

i and D2i if there

is a connected subset ci sub c satisfying f(ci) sub Di andf(ci)capD1

i and f(ci)capD2i are nonempty In this case it is

denoted by f(c)⟼Di

Table 1 Equilibrium points and eigenvalues of system (1)

Equilibrium point Eigenvalues Type of equilibrium pointP0 (0 0 0) λ1 minus 1 λ2 3 λ3 0 Nonhyperbolic

P1 (06847 01141 025)λ1 11087 + 06279iλ2 11087 ndash 06279i

λ3 minus 11549Hyperbolic

P2 (ndash06847 -01141 025)λ1 11087 + 06279iλ211087 ndash 06279i


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Time (s)

ndash05

0

50

1

λ

λ1λ2

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Time (s)

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ3

(b)

Figure 1 Lyapunov exponents of system (1) when a 1 b 6 c 5 d 1 e 3 and f 2

Complexity 3

Definition 3 Suppose that σ Σm⟶Σm and f S⟶ S

are both continuous functions with topological spaces Σmand S respectively If there exists a continuous surjectionh Σm⟶ S confirming to f∘h h∘σ it is said that f istopologically semiconjugate to function σ

Lemma 1 If fp(Di)⟼Di then we have fmp(Di)⟼Diwhere m is a positive integer

Lemma 2 If fp(D1)⟼D1 fp(D1)⟼D2 fq(D2)

⟼D1 and fq(D2)⟼D2 then there would exist acompact invariant set M sub D such that fp+q | M is semi-conjugate to 2-shift dynamics and the topological entropy of f

will satisfy ent(f)ge [1(p + q)]log 2

32 Finding Topological Horseshoe in the Dynamical SystemAccording to the theory above a horseshoe will be found inthe dynamical system by three steps [17] In this process weset the parameters of system (1) as a 1 b 6 c 5 d 1e 3 f 2 and initial condition x (0) (001 001 005)

Step 1 As shown in Figure 6 we first select four vertices ofthe Poincare section P on the planeΘ (x1 x2 x3) isin R3 x2 01113864 1113865 as (ndash8 0 ndash05) (ndash8 0 15)(8 0 15) and (8 0 ndash05)

Step 2 en after many trial-and-error numerical simu-lations we carefully pick a quadrilateral D1 of quadrangle Pwith the four vertices being

(minus 7618610595 0 1062250623)

(minus 7643703532 0 1049345387)

(minus 7239428439 0 1018765586)

(minus 7215729554 0 1031951372)

(4)

Let us suppose that D11 denotes the left side while D2

1denotes the right side of quadrilateralD1 It is known from thenumerical result that the third return map H3(D1

1) lies on theleft side of D1 but the third return map H3(D2

1) lies on theright side of D1 ereby under this return map the imageH3(x) (x isin D1) lies wholly across the quadrangle D1 withrespect to the sides D1

1 and D21 seen in Figure 7(a)

minus10 minus5 0 5 10minus6

minus4

minus2

0

2

4

6

x2

x1

(a)

x3

x2minus6 minus4 minus2 0 2 4 6

minus1

minus05

0

05

1

15

2

(b)

x3

x1minus10 minus5 0 5 10

minus1

minus05

0

05

1

15

2

(c)

Figure 2 Chaotic phase diagrams of the proposed system

4 Complexity

ndash8 ndash6 ndash4 ndash2ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

0 2 4 6 8x1

x2

(a)

ndash3 ndash2 ndash1ndash075

ndash07

ndash065

ndash06

ndash055

ndash05

ndash045

ndash04

ndash035

ndash03

ndash025

0 1 2 3x2

x3

(b)

ndash10 ndash5 0 5 10ndash1

ndash05

0

05

1

15

x1

x3

(c)

Figure 3 Poincare maps on the plane of (a) x3 0 (b) x1 0 and (c) x2 0

c

06

08

1

12

14

16

18

2

22

24

26

2 3 4 5 6 7 8

x3

(a)

2 3 4 5 6 7c

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)

Figure 4 (a) Bifurcation diagram and (b) Lyapunov exponents of system (1) versus parameter c

Complexity 5

Step 3 Finally we will take another quadrilateral D2 of P suchthat H(D1)⟼D2 H(D2)⟼D2 and H(D2)⟼D1 By agreat deal of attempts the four vertexes ofD2 are picked as follows

(minus 7865357807 0 1082029302)

(minus 7896026952 0 1066318579)

(minus 7661826208 0 1048363466)

(minus 7633945167 0 1063232544)

(5)

Analogously D12 and D2

2 indicate the left and right sides ofquadrilateral D2 respectively e numerical simulations ofthe third return Poincare map H3(D2) and the enlarged vieware depicted in Figures 7(b) and 7(c) respectively It is shownfrom the figures that the return map H3(x) (x isin D2) suitablyacross the quadrangles D1 and D2 with H3(D1

2) lying on theright side of D1 and H3(D2

2) lying on the left side of D2

erefore we conclude by virtue of Lemma 2 that therewould exist a compact invariant setM sub D such thatH6 | M issemiconjugate to 2-shift dynamics and we obtainent(H)ge (log 2)6gt 0us system (1) is proved to be chaoticin theory with the parameters of a 1 b 6 c 5 d 1 e 3f 2 and the initial condition of x (0) (001 001 005)

4 Stabilization for the Dynamical System

41 Control Scheme In order to stabilize the proposeddynamical system we add the single controller u on thesecond equation us the controlled dynamical system isdepicted as follows

_x1 minus ax1 + bx2

_x2 ex2 minus fx1x3 + u


⎧⎪⎪⎨

⎪⎪⎩(6)

e purpose of our design is to propose suitable controlscheme u such that all the output variables of system (6)converge to the zero equilibrium point asymptotically

Theorem 1 For the controlled system (6) we design thesingle controller u as

u Ksign x2( 1113857 (7)

If the control gain satisfies Kle minus ((b24a) + e)B2 withB2 ge x2 then the output variables of the controlled system(6) converges to the zero equilibrium point asymptotically

Proof We choose the candidate Lyapunov function as

V 05x21 + x2

2 + fx23

d1113888 1113889 (8)

e corresponding time derivative of V(x) is deduced by

1 15 2 25 3f

08

1

12

14

16

18

2

x3

(a)

1 12 14 16 18 2 22 24 26 28 3f

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)

Figure 5 (a) Bifurcation diagram and (b) Lyapunov exponents of system (1) versus parameter f

minus10minus5

05

10

minus5

0

5minus1

minus05

0

05

1

15

2

x3

x2x1

Figure 6 Poincare cross section of the proposed system

6 Complexity

_V x1 _x1 + x2 _x2 + fx3 _x3

d

minus ax21 + bx1x2 + ex

22 minus fx1x2x3

+ Kx2sign x2( 1113857 minuscfx4

3d

+ fx1x2x3


22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

minus ax21 minus bx1x2 +

b

2a

radic x21113888 1113889

2⎡⎣ ⎤⎦ +

b

2a

radic x21113888 1113889

2

+ ex22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889 x2

11138681113868111386811138681113868111386811138681113868 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

(9)

us when Kle minus ((b24a) + e)B2 we find that

_Vle minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889B2 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

le minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

minuscfx4

3dle 0

(10)

erefore the output variables of the controlled system(6) will converge to the zero equilibrium pointasymptotically

42 Numerical Verification In this section numericalsimulations are executed by adopting the ODE45 method toverify the availability of the proposed control scheme Forcomparing conveniently we choose the system parametersas a 1 b 6 c 5 d 1 e 3 and f 2 and the initial statesare set as x (0) (001 001 001) With the parameter setsystem (6) is chaotic before the controller is put into effect

minus8 minus78 minus76 minus74 minus72 minus7 minus68 minus66 minus64094

096

098

1

102

104

106

108

11H3 (D1

1)

H3 (D21)

D11

D21

D1

x1

x3

(a)

H3 (D22)

H3 (D12)

H3 (D21)

D22

D12

D1

D2

minus85 minus8 minus75 minus7 minus65 minus6 minus55 minus5 minus45 minus407

075

08

085

09

095

1

105

11

115

x1

x3

(b)

H3 (D22)

D12

D22

D1

D2

minus82 minus8 minus78 minus76 minus74 minus72 minus71

101

102

103

104

105

106

107

108

109

11

x1

x3

(c)

Figure 7 (a) e subset D1 and its image (b) and (c) the subset D2 and its image

Complexity 7

e controller u is exerted at 60th second and thecontrol gain is taken as K ndash2 and K ndash16 Figures 8(a) and8(b) depict the control results from which one can see thatwe can obtain good control effect with only taking a shorttime for the system to be stabilized at the fixed point And aswe can also know only small control energy is needed toreach our control purpose

5 DSP-Based Realization of the Attractors andStabilization Scheme

e implementation of continuous chaotic systems withelectronic circuit has been widely adopted e parametertolerance of the components of analog electronic circuitwill cause the trajectory change of the chaotic systemwhich restricts its practical engineering applicationerefore DSP-based implement of continuous chaoticsystem can overcome these problems effectively In thissection the attractors of system (1) and stabilizationscheme for system (6) are implemented based on the DSPplatform In our experiments the Texas Instrument DSPTMS320F28335 is employed to calculate the state variableand the control variable which can run at 150MHz andinterfaces with a 12-bit quad-channel digital-to-analogconverter DAC7724 by parallel bus (PB) mode Controlsignals required by DAC7724 are generated by DSP andCPLD chips e block diagram of hardware platform isshown in Figure 9

In practice we discretize the continuous system by theclassical fourth-order RungendashKutta algorithm with thesampling period ΔT From Figure 2 we know that theamplitude of variable x3 (|x3|lt 2) is too small to affect thecomputational precision erefore we proposed a generalmethod to rescale the system variables by scaling factors kiwhere i 1 2 3

Setting ui kixi i 1 2 3 system (1) can be trans-formed to

_ui fi a b c d e f k1 k2 k3 u1 u2u31113872 1113873 (11)

where i 1 2 3When introducing the linear transformation of

ui⟶ xi the corresponding differential equation of system(11) can be expressed as

_x1 minus ax1 +bk1

k2x2

_x2 ex2 minusfk2

k1k3x1x3

_x3 minusc

k23x33 +

dk3

k1k2x1x2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

Discretizing system (12) using the classical fourth-orderRungendashKutta algorithm the following difference equationscan be obtained

xi(n + 1) xi(n) + ΔTKi1 + 2Ki2 + 2Ki3 + Ki41113872 1113873

6 (13)

where i 1 2 3 and

ndash10

0

10

x1

x2

x3

ndash5

0

5

Applying control

Time (s)

ndash1

0

1

2

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

(a)

x1

x2

x3

ndash10

0

10

ndash5

0

5

Applying control

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100Time (s)

ndash1

0

1

2

(b)

Figure 8 Control results of state variables with (a) K ndash2 and(b) K ndash16

DSP(TMS320F28335)

DA converter(DAC7724)

OSC(Tektronix TDS 2014B)

PB

CPLD(EPM1270T144C5N)

XY

Z

Figure 9 Block diagram for DSP implementation of chaoticattractors and its stabilization

8 Complexity

K11 minus ax1(n) +bk1

k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

K1j minus a x1(n) + mK1jminus 11113960 1113961 +bk1k2

x2(n) + mK2jminus 11113960 1113961

K2j e x2(n) + mK2jminus 11113960 1113961 minusfk2

k1k3x1(n) + mK1jminus 11113960 1113961 x3(n) + mK3jminus 11113960 1113961

K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)

Figure 10 DSP-based realization for (a)ndash(c) chaotic attractors (d) stabilization process (e) experiment platform

Complexity 9

When j 2 3 m 05 and when j 4 m 1e digital sequences x1(n) x2(n) x3(n) are produced by

DSP according to (13)ndash(15) and converted into the corre-sponding analog sequences by the DAC7724 and then trans-mitted to oscilloscope With the set a 1 b 6 c 5 d 1e 3 f 2 k1 1 k2 1 k3 2 x (0) (001 001 001) andthe control gain K ndash6 the results of DSP-based realization ofthe attractor and stabilization scheme are depicted in Figure 10By comparing with the attractors in Figure 2 and stabilizationprocess in Figure 7 simulated by Matlab it can be concludedthat they have a good qualitative agreement

6 Conclusion

In this paper we presented a three-dimensional dynamicalsystem with a simple algebraic structure Basic dynamicalproperties of the system including equilibrium point phaseportrait Poincare map parameter bifurcation and Lyapu-nov exponent are studied through theoretical analysis andnumerical simulation And the theory of topologicalhorseshoe is adopted to rigorously prove the chaoticemergence of the system theoretically en based on theLyapunov stability criterion we designed a single controlscheme to stabilize the chaotic system to its zero-equilibriumpoint e attractor and stabilization process are realized viaDSP-based technology which have a good qualitativeagreement to the Matlab simulation thus it well confirmedthe validity and enforceability of the theoretical scheme

Data Availability

e figure data and table data used to support the findings ofthis study are included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China under Grant nos 51977153 and51577046 the State Key Program of National Natural Sci-ence Foundation of China under Grant no 51637004 theNational Key Research and Development Plan ldquoImportantScientific Instruments and Equipment Developmentrdquo underGrant no 2016YFF0102200 the Equipment Research Projectin Advance under Grant no 41402040301 the ResearchFoundation of Education Bureau of Hunan Province ofChina under Grant no 18A314 Hunan Provincial NaturalScience Foundation of China under Grant no 2019JJ40109and the Science and Research Creative Team of HunanInstitute of Science and Technology (no 2019-TD-10)

References

[1] J Mou K Sun J Ruan and S He ldquoA nonlinear circuit withtwo memcapacitorsrdquo Nonlinear Dynamics vol 86 no 3pp 1735ndash1744 2016

[2] J Zhou X Wang D Xu and S Bishop ldquoNonlinear dynamiccharacteristics of a quasi-zero stiffness vibration isolator withcam-roller-spring mechanismsrdquo Journal of Sound and Vi-bration vol 346 pp 53ndash69 2015

[3] F Peng Q Long Z-X Lin and M Long ldquoA reversiblewatermarking for authenticating 2D CAD engineeringgraphics based on iterative embedding and virtual coordi-natesrdquo Multimedia Tools and Applications vol 78 no 19pp 26885ndash26905 2017

[4] Z Wei Y Li B Sang Y Liu and W Zhang ldquoComplexdynamical behaviors in a 3D simple chaotic flow with 3Dstable or 3D unstable manifolds of a single equilibriumrdquoInternational Journal of Bifurcation and Chaos vol 29 no 7p 1950095 2019

[5] Y Zhou Z Hua C M Pun and C L Chen ldquoCascade chaoticsystem with applicationsrdquo IEEE Transactions on Cyberneticsvol 45 no 45 pp 2001ndash2012 2015

[6] M Zapateiro Y Vidal and L Acho ldquoA secure communi-cation scheme based on chaotic Duffing oscillators and fre-quency estimation for the transmission of binary-codedmessagesrdquo Communications in Nonlinear Science and Nu-merical Simulation vol 19 no 4 pp 991ndash1003 2014

[7] C Zhu S Xu Y Hu and K Sun ldquoBreaking a novel imageencryption scheme based on Brownian motion and PWLCMchaotic systemrdquo Nonlinear Dynamics vol 79 no 2pp 1511ndash1518 2015

[8] G Ye C Pan X Huang Z Zhao and J He ldquoA chaotic imageencryption algorithm based on information entropyrdquo Inter-national Journal of Bifurcation and Chaos vol 28 no 1Article ID 1850010 2018

[9] X Xiang and B Shi ldquoWeak signal detection based on theinformation fusion and chaotic oscillatorrdquo Chaos vol 20no 1 Article ID 013104 2010

[10] Y Ding K Sun and X Xu ldquoHough-MHAF localizationalgorithm for dual-frequency continuous-wave through-wallradarrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 52 no 1 pp 111ndash121 2016

[11] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal ofAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[12] J Kennedy S Koccedilak and J A YORKE ldquoA chaos lemmardquo2eAmerican Mathematical Monthly vol 108 no 5 pp 411ndash4232001

[13] X-S Yang and Y Tang ldquoHorseshoes in piecewise continuousmapsrdquo Chaos Solitons amp Fractals vol 19 no 4 pp 841ndash8452004

[14] X-S Yang ldquoTopological horseshoes and computer assistedverification of chaotic dynamicsfication of chaotic dynamicsrdquoInternational Journal of Bifurcation and Chaos vol 19 no 4pp 1127ndash1145 2009

[15] W-Z Huang and Y Huang ldquoChaos bifurcation and ro-bustness of a class of hopfield neural networksrdquo InternationalJournal of Bifurcation and Chaos vol 21 no 3 pp 885ndash8952011

[16] C Ma and X Wang ldquoHopf bifurcation and topologicalhorseshoe of a novel finance chaotic systemrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 17no 2 pp 721ndash730 2012

[17] Q Li and X-S Yang ldquoA simple method for finding topo-logical horseshoesrdquo International Journal of Bifurcation andChaos vol 20 no 2 pp 467ndash478 2010

[18] C Li L Wu H Li and Y Tong ldquoA novel chaotic system andits topological horseshoerdquo Nonlinear Analysis Modelling andControl vol 18 no 1 pp 66ndash77 2013

10 Complexity

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

[20] K Pyragas ldquoContinuous control of chaos by self-controllingfeedbackrdquo Physics Letters A vol 170 no 6 pp 421ndash428 1992

[21] F Farivar M A Nekoui M A Shoorehdeli andM Teshnehlab ldquoModified projective synchronization ofchaotic dissipative gyroscope systems via backstepping con-trolrdquo Indian Journal of Physics vol 86 no 10 pp 901ndash9062012

[22] C Li K Su and L Wu ldquoAdaptive sliding mode control forsynchronization of a fractional-order chaotic systemrdquo Journalof Computer Nonlinear Dynamics vol 8 no 3 Article ID031005 2013

[23] E Fridman ldquoA refined input delay approach to sampled-datacontrolrdquo Automatica vol 46 no 2 pp 421ndash427 2010

[24] A B MUZAFFAR and K AYUB ldquoMultiswitching combi-nation synchronization of non-identical fractional-orderchaotic systemsrdquo Pramana-J Physvol 90 no 6 2018

[25] Z Wei J C Sprott and H Chen ldquoElementary quadraticchaotic flows with a single non-hyperbolic equilibriumrdquoPhysics Letters A vol 379 no 37 pp 2184ndash2187 2015

[26] A T Azar C Volos N A Gerodimos et al ldquoA novel chaoticsystem without equilibrium dynamics synchronization andcircuit realizationrdquo Complexity vol 2017 no 11 Article ID7871467 2017

[27] G Y Peng F H Min and E R Wang ldquoCircuit imple-mentation synchronization of multistability and image en-cryption of a four-wingmemristive chaotic systemrdquo Journal ofElectrical and Computer Engineering vol 2018 Article ID8649294 13 pages 2018

Complexity 11

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

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Submit your manuscripts atwwwhindawicom

method is restricted by the level of noise in the experimentaldata for the sake of the discrete nature of the control signalen a continuous delayed feedback scheme is proposedconsequently [20] e control signal is the perturbations ofthe difference between the states of the current system andone period of the target orbit in the past erefore theintensity of the perturbations will vanish when the systemevolves to the desired orbit Since then a wide variety ofcontrol approaches emerged for different applications suchas backstepping control [21] sliding mode control [22]sampled-data control [23] and multiswitching combinationcontrol [24] just to name a few However the presentschemes focused on the control problem need severalcontrollers From the points of both practical applicationand theoretical research it is significant to design simple yetexecutable control technique

In this paper an autonomous chaotic system with asimple algebraic structure of six terms is proposed Basicdynamical properties of the system including equilibriumpoint phase portrait Poincare map parameter bifurcationand Lyapunov exponent are studied in theory and nu-merical simulation It is found that this system exhibitsfruitful dynamic behaviors of dense periodic windows andcoexistence of nonhyperbolic and hyperbolic equilibriumpoints And to rigorously verify the emergence of chaos ofthis system in theory the topological horseshoe is investi-gated in the phase space of Poincare mapen based on theLyapunov stability criterion a single control scheme isdesigned to stabilize the chaotic system to its zero-equi-librium point e implementation scheme of attractors andcontrol scheme are discussed in detail and realized via DSP-based technique confirming the validity and enforceabilityof the theoretical scheme

2 The Proposed Dynamical System

e autonomous dynamical system considered here is givenby

_x1 minus ax1 + bx2

_x2 ex2 minus fx1x3


⎧⎪⎪⎨

⎪⎪⎩(1)

Differential equations of system (1) are simple with acubic term and two quadratic cross-product terms esystem parameters a b c d e and f are all positiveconstants

21DissipativityandExistenceofAttractor We first considerthe general condition of dissipativity to ensure the chaoticproperty

nablaV z _x1

x1+

z _x2

x2+

z _x3

x3 minus a + e minus 3cx

23 (2)

erefore system (1) would be dissipative and willconverge to a subset of measure zero volume according todVdt e(minus a+eminus 3cx2

3) when satisfying minus a + e minus 3cx23 lt 0 is

means that the volume will become V(0)e(minus a+eminus 3cx23)t at time

t through the flow generated by the system for an initialvolume V(0) erefore there exists an attractor in system(1) with minus a + e minus 3cx2

3 lt 0

22 Equilibrium Points and Stability Considering the con-dition of equilibrium point _x1 0 _x2 0 and _x3 0 weobtain three equilibrium points of system (1) as follows

P0(0 0 0)

P1a

b

ce3

df3

1113971

a2

b2

ce3

df3

1113971

ae

bf⎛⎝ ⎞⎠

P2 minusa

b

ce3

df3

1113971

minusa2

b2

ce3

df3

1113971

ae

bf⎛⎝ ⎞⎠

(3)

When selecting a 1 b 6 c 5 d 1 e 3 and f 2 theequilibrium points and the corresponding eigenvalues areshown in Table 1 including the type of equilibrium points

From Table 1 it is known that the three equilibriumpoints are all unstable with stable manifold and unstablemanifold According to [25] since the characteristic value λ3of equilibrium point P0 equals to zero the equilibrium pointis nonhyperbolic type However equilibrium points P1 andP2 are hyperbolic since all the real parts of the correspondingeigenvalues of these two equilibrium points are nonzeroerefore this is a chaotic system in which hyperbolic andnonhyperbolic equilibrium points coexist And this kind ofchaotic system does not belong to Silrsquonikov sense of thechaotic system and it is difficult to prove the existence ofchaos by Silrsquonikov theorem At present chaotic systems withspecial features such as chaotic systems with nonequilib-rium with multistability and with hyperbolic and non-hyperbolic equilibrium coexisting have attracted extensiveattention by researchers [26 27]

23 Phase Portrait and Chaotic Properties When selectinga 1 b 6 c 5 d 1 e 3 f 2 and initial values x(0) (001 001 005) the Lyapunov exponents of system (1)are depicted in Figures 1(a) and 1(b) with the values as0123017 0000029 and minus 11414797 According to the threeLyapunov exponents the KaplanndashYorke dimension isDKY 2 + (0123017 + 0000029)11414797 20108 ere-fore the KaplanndashYorke dimension is fractional e cor-responding chaotic phase diagrams further reveal that theproposed system displays complicated chaotic behaviors asdepicted in Figure 2

As an important analytical technique Poincare map canreflect the bifurcation and folding properties of chaosSelecting the parameter values a 1 b 6 c 5 d 1 e 3and f 2 and taking the crossing planes x3 0 x1 0 andx2 0 we obtain the corresponding Poincare maps illus-trated in Figure 3 We can see the attractor structure fromthe dense dots

24 Influence of Parameter Variation It is found that system(1) exhibits complicated dynamical behaviors with the

2 Complexity











iharrc

D2i


i and D2i if there






P1 (06847 01141 025)λ1 11087 + 06279iλ2 11087 ndash 06279i




0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Time (s)

ndash05

0

50

1

λ

λ1λ2

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Time (s)

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ3

(b)


Complexity 3










(minus 7618610595 0 1062250623)

(minus 7643703532 0 1049345387)

(minus 7239428439 0 1018765586)

(minus 7215729554 0 1031951372)

(4)







minus4

minus2

0

2

4

6

x2

x1

(a)

x3


minus1

minus05

0

05

1

15

2

(b)

x3


minus1

minus05

0

05

1

15

2

(c)


4 Complexity


ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

0 2 4 6 8x1

x2

(a)


ndash07

ndash065

ndash06

ndash055

ndash05

ndash045

ndash04

ndash035

ndash03

ndash025

0 1 2 3x2

x3

(b)


ndash05

0

05

1

15

x1

x3

(c)


c

06

08

1

12

14

16

18

2

22

24

26

2 3 4 5 6 7 8

x3

(a)

2 3 4 5 6 7c

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


Complexity 5


(minus 7865357807 0 1082029302)

(minus 7896026952 0 1066318579)

(minus 7661826208 0 1048363466)

(minus 7633945167 0 1063232544)

(5)








_x1 minus ax1 + bx2



⎧⎪⎪⎨

⎪⎪⎩(6)



u Ksign x2( 1113857 (7)



V 05x21 + x2

2 + fx23

d1113888 1113889 (8)


1 15 2 25 3f

08

1

12

14

16

18

2

x3

(a)

1 12 14 16 18 2 22 24 26 28 3f

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


minus10minus5

05

10

minus5

0

5minus1

minus05

0

05

1

15

2

x3

x2x1


6 Complexity

_V x1 _x1 + x2 _x2 + fx3 _x3

d


22 minus fx1x2x3


3d

+ fx1x2x3


22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d


b

2a

radic x21113888 1113889

2⎡⎣ ⎤⎦ +

b

2a

radic x21113888 1113889

2

+ ex22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889 x2

11138681113868111386811138681113868111386811138681113868 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

(9)


_Vle minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889B2 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

le minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

minuscfx4

3dle 0

(10)




096

098

1

102

104

106

108

11H3 (D1

1)

H3 (D21)

D11

D21

D1

x1

x3

(a)

H3 (D22)

H3 (D12)

H3 (D21)

D22

D12

D1

D2


075

08

085

09

095

1

105

11

115

x1

x3

(b)

H3 (D22)

D12

D22

D1

D2


101

102

103

104

105

106

107

108

109

11

x1

x3

(c)


Complexity 7









_x1 minus ax1 +bk1

k2x2

_x2 ex2 minusfk2

k1k3x1x3

_x3 minusc

k23x33 +

dk3

k1k2x1x2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



6 (13)

where i 1 2 3 and

ndash10

0

10

x1

x2

x3

ndash5

0

5

Applying control

Time (s)

ndash1

0

1

2

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

(a)

x1

x2

x3

ndash10

0

10

ndash5

0

5

Applying control

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100Time (s)

ndash1

0

1

2

(b)


DSP(TMS320F28335)



PB


XY

Z


8 Complexity


k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


x2(n) + mK2jminus 11113960 1113961



K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)


Complexity 9



6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018


















iharrc

D2i


i and D2i if there






P1 (06847 01141 025)λ1 11087 + 06279iλ2 11087 ndash 06279i




0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Time (s)

ndash05

0

50

1

λ

λ1λ2

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Time (s)

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ3

(b)


Complexity 3










(minus 7618610595 0 1062250623)

(minus 7643703532 0 1049345387)

(minus 7239428439 0 1018765586)

(minus 7215729554 0 1031951372)

(4)







minus4

minus2

0

2

4

6

x2

x1

(a)

x3


minus1

minus05

0

05

1

15

2

(b)

x3


minus1

minus05

0

05

1

15

2

(c)


4 Complexity


ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

0 2 4 6 8x1

x2

(a)


ndash07

ndash065

ndash06

ndash055

ndash05

ndash045

ndash04

ndash035

ndash03

ndash025

0 1 2 3x2

x3

(b)


ndash05

0

05

1

15

x1

x3

(c)


c

06

08

1

12

14

16

18

2

22

24

26

2 3 4 5 6 7 8

x3

(a)

2 3 4 5 6 7c

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


Complexity 5


(minus 7865357807 0 1082029302)

(minus 7896026952 0 1066318579)

(minus 7661826208 0 1048363466)

(minus 7633945167 0 1063232544)

(5)








_x1 minus ax1 + bx2



⎧⎪⎪⎨

⎪⎪⎩(6)



u Ksign x2( 1113857 (7)



V 05x21 + x2

2 + fx23

d1113888 1113889 (8)


1 15 2 25 3f

08

1

12

14

16

18

2

x3

(a)

1 12 14 16 18 2 22 24 26 28 3f

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


minus10minus5

05

10

minus5

0

5minus1

minus05

0

05

1

15

2

x3

x2x1


6 Complexity

_V x1 _x1 + x2 _x2 + fx3 _x3

d


22 minus fx1x2x3


3d

+ fx1x2x3


22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d


b

2a

radic x21113888 1113889

2⎡⎣ ⎤⎦ +

b

2a

radic x21113888 1113889

2

+ ex22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889 x2

11138681113868111386811138681113868111386811138681113868 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

(9)


_Vle minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889B2 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

le minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

minuscfx4

3dle 0

(10)




096

098

1

102

104

106

108

11H3 (D1

1)

H3 (D21)

D11

D21

D1

x1

x3

(a)

H3 (D22)

H3 (D12)

H3 (D21)

D22

D12

D1

D2


075

08

085

09

095

1

105

11

115

x1

x3

(b)

H3 (D22)

D12

D22

D1

D2


101

102

103

104

105

106

107

108

109

11

x1

x3

(c)


Complexity 7









_x1 minus ax1 +bk1

k2x2

_x2 ex2 minusfk2

k1k3x1x3

_x3 minusc

k23x33 +

dk3

k1k2x1x2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



6 (13)

where i 1 2 3 and

ndash10

0

10

x1

x2

x3

ndash5

0

5

Applying control

Time (s)

ndash1

0

1

2

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

(a)

x1

x2

x3

ndash10

0

10

ndash5

0

5

Applying control

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100Time (s)

ndash1

0

1

2

(b)


DSP(TMS320F28335)



PB


XY

Z


8 Complexity


k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


x2(n) + mK2jminus 11113960 1113961



K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)


Complexity 9



6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018

















(minus 7618610595 0 1062250623)

(minus 7643703532 0 1049345387)

(minus 7239428439 0 1018765586)

(minus 7215729554 0 1031951372)

(4)







minus4

minus2

0

2

4

6

x2

x1

(a)

x3


minus1

minus05

0

05

1

15

2

(b)

x3


minus1

minus05

0

05

1

15

2

(c)


4 Complexity


ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

0 2 4 6 8x1

x2

(a)


ndash07

ndash065

ndash06

ndash055

ndash05

ndash045

ndash04

ndash035

ndash03

ndash025

0 1 2 3x2

x3

(b)


ndash05

0

05

1

15

x1

x3

(c)


c

06

08

1

12

14

16

18

2

22

24

26

2 3 4 5 6 7 8

x3

(a)

2 3 4 5 6 7c

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


Complexity 5


(minus 7865357807 0 1082029302)

(minus 7896026952 0 1066318579)

(minus 7661826208 0 1048363466)

(minus 7633945167 0 1063232544)

(5)








_x1 minus ax1 + bx2



⎧⎪⎪⎨

⎪⎪⎩(6)



u Ksign x2( 1113857 (7)



V 05x21 + x2

2 + fx23

d1113888 1113889 (8)


1 15 2 25 3f

08

1

12

14

16

18

2

x3

(a)

1 12 14 16 18 2 22 24 26 28 3f

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


minus10minus5

05

10

minus5

0

5minus1

minus05

0

05

1

15

2

x3

x2x1


6 Complexity

_V x1 _x1 + x2 _x2 + fx3 _x3

d


22 minus fx1x2x3


3d

+ fx1x2x3


22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d


b

2a

radic x21113888 1113889

2⎡⎣ ⎤⎦ +

b

2a

radic x21113888 1113889

2

+ ex22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889 x2

11138681113868111386811138681113868111386811138681113868 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

(9)


_Vle minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889B2 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

le minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

minuscfx4

3dle 0

(10)




096

098

1

102

104

106

108

11H3 (D1

1)

H3 (D21)

D11

D21

D1

x1

x3

(a)

H3 (D22)

H3 (D12)

H3 (D21)

D22

D12

D1

D2


075

08

085

09

095

1

105

11

115

x1

x3

(b)

H3 (D22)

D12

D22

D1

D2


101

102

103

104

105

106

107

108

109

11

x1

x3

(c)


Complexity 7









_x1 minus ax1 +bk1

k2x2

_x2 ex2 minusfk2

k1k3x1x3

_x3 minusc

k23x33 +

dk3

k1k2x1x2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



6 (13)

where i 1 2 3 and

ndash10

0

10

x1

x2

x3

ndash5

0

5

Applying control

Time (s)

ndash1

0

1

2

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

(a)

x1

x2

x3

ndash10

0

10

ndash5

0

5

Applying control

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100Time (s)

ndash1

0

1

2

(b)


DSP(TMS320F28335)



PB


XY

Z


8 Complexity


k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


x2(n) + mK2jminus 11113960 1113961



K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)


Complexity 9



6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018









ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

0 2 4 6 8x1

x2

(a)


ndash07

ndash065

ndash06

ndash055

ndash05

ndash045

ndash04

ndash035

ndash03

ndash025

0 1 2 3x2

x3

(b)


ndash05

0

05

1

15

x1

x3

(c)


c

06

08

1

12

14

16

18

2

22

24

26

2 3 4 5 6 7 8

x3

(a)

2 3 4 5 6 7c

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


Complexity 5


(minus 7865357807 0 1082029302)

(minus 7896026952 0 1066318579)

(minus 7661826208 0 1048363466)

(minus 7633945167 0 1063232544)

(5)








_x1 minus ax1 + bx2



⎧⎪⎪⎨

⎪⎪⎩(6)



u Ksign x2( 1113857 (7)



V 05x21 + x2

2 + fx23

d1113888 1113889 (8)


1 15 2 25 3f

08

1

12

14

16

18

2

x3

(a)

1 12 14 16 18 2 22 24 26 28 3f

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


minus10minus5

05

10

minus5

0

5minus1

minus05

0

05

1

15

2

x3

x2x1


6 Complexity

_V x1 _x1 + x2 _x2 + fx3 _x3

d


22 minus fx1x2x3


3d

+ fx1x2x3


22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d


b

2a

radic x21113888 1113889

2⎡⎣ ⎤⎦ +

b

2a

radic x21113888 1113889

2

+ ex22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889 x2

11138681113868111386811138681113868111386811138681113868 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

(9)


_Vle minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889B2 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

le minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

minuscfx4

3dle 0

(10)




096

098

1

102

104

106

108

11H3 (D1

1)

H3 (D21)

D11

D21

D1

x1

x3

(a)

H3 (D22)

H3 (D12)

H3 (D21)

D22

D12

D1

D2


075

08

085

09

095

1

105

11

115

x1

x3

(b)

H3 (D22)

D12

D22

D1

D2


101

102

103

104

105

106

107

108

109

11

x1

x3

(c)


Complexity 7









_x1 minus ax1 +bk1

k2x2

_x2 ex2 minusfk2

k1k3x1x3

_x3 minusc

k23x33 +

dk3

k1k2x1x2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



6 (13)

where i 1 2 3 and

ndash10

0

10

x1

x2

x3

ndash5

0

5

Applying control

Time (s)

ndash1

0

1

2

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

(a)

x1

x2

x3

ndash10

0

10

ndash5

0

5

Applying control

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100Time (s)

ndash1

0

1

2

(b)


DSP(TMS320F28335)



PB


XY

Z


8 Complexity


k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


x2(n) + mK2jminus 11113960 1113961



K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)


Complexity 9



6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018









(minus 7865357807 0 1082029302)

(minus 7896026952 0 1066318579)

(minus 7661826208 0 1048363466)

(minus 7633945167 0 1063232544)

(5)








_x1 minus ax1 + bx2



⎧⎪⎪⎨

⎪⎪⎩(6)



u Ksign x2( 1113857 (7)



V 05x21 + x2

2 + fx23

d1113888 1113889 (8)


1 15 2 25 3f

08

1

12

14

16

18

2

x3

(a)

1 12 14 16 18 2 22 24 26 28 3f

ndash12

ndash10

ndash8

ndash6

ndash4

ndash2

0

λ

λ1λ2λ3

(b)


minus10minus5

05

10

minus5

0

5minus1

minus05

0

05

1

15

2

x3

x2x1


6 Complexity

_V x1 _x1 + x2 _x2 + fx3 _x3

d


22 minus fx1x2x3


3d

+ fx1x2x3


22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d


b

2a

radic x21113888 1113889

2⎡⎣ ⎤⎦ +

b

2a

radic x21113888 1113889

2

+ ex22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889 x2

11138681113868111386811138681113868111386811138681113868 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

(9)


_Vle minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889B2 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

le minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

minuscfx4

3dle 0

(10)




096

098

1

102

104

106

108

11H3 (D1

1)

H3 (D21)

D11

D21

D1

x1

x3

(a)

H3 (D22)

H3 (D12)

H3 (D21)

D22

D12

D1

D2


075

08

085

09

095

1

105

11

115

x1

x3

(b)

H3 (D22)

D12

D22

D1

D2


101

102

103

104

105

106

107

108

109

11

x1

x3

(c)


Complexity 7









_x1 minus ax1 +bk1

k2x2

_x2 ex2 minusfk2

k1k3x1x3

_x3 minusc

k23x33 +

dk3

k1k2x1x2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



6 (13)

where i 1 2 3 and

ndash10

0

10

x1

x2

x3

ndash5

0

5

Applying control

Time (s)

ndash1

0

1

2

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

(a)

x1

x2

x3

ndash10

0

10

ndash5

0

5

Applying control

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100Time (s)

ndash1

0

1

2

(b)


DSP(TMS320F28335)



PB


XY

Z


8 Complexity


k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


x2(n) + mK2jminus 11113960 1113961



K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)


Complexity 9



6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018








_V x1 _x1 + x2 _x2 + fx3 _x3

d


22 minus fx1x2x3


3d

+ fx1x2x3


22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d


b

2a

radic x21113888 1113889

2⎡⎣ ⎤⎦ +

b

2a

radic x21113888 1113889

2

+ ex22 + K x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889 x2

11138681113868111386811138681113868111386811138681113868 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

(9)


_Vle minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

+b2

4a+ e1113888 1113889B2 + K1113890 1113891 x2

11138681113868111386811138681113868111386811138681113868 minus

cfx43

d

le minusa

radicx1 minus

b

2a

radic x21113890 1113891

2

minuscfx4

3dle 0

(10)




096

098

1

102

104

106

108

11H3 (D1

1)

H3 (D21)

D11

D21

D1

x1

x3

(a)

H3 (D22)

H3 (D12)

H3 (D21)

D22

D12

D1

D2


075

08

085

09

095

1

105

11

115

x1

x3

(b)

H3 (D22)

D12

D22

D1

D2


101

102

103

104

105

106

107

108

109

11

x1

x3

(c)


Complexity 7









_x1 minus ax1 +bk1

k2x2

_x2 ex2 minusfk2

k1k3x1x3

_x3 minusc

k23x33 +

dk3

k1k2x1x2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



6 (13)

where i 1 2 3 and

ndash10

0

10

x1

x2

x3

ndash5

0

5

Applying control

Time (s)

ndash1

0

1

2

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

(a)

x1

x2

x3

ndash10

0

10

ndash5

0

5

Applying control

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100Time (s)

ndash1

0

1

2

(b)


DSP(TMS320F28335)



PB


XY

Z


8 Complexity


k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


x2(n) + mK2jminus 11113960 1113961



K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)


Complexity 9



6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018
















_x1 minus ax1 +bk1

k2x2

_x2 ex2 minusfk2

k1k3x1x3

_x3 minusc

k23x33 +

dk3

k1k2x1x2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



6 (13)

where i 1 2 3 and

ndash10

0

10

x1

x2

x3

ndash5

0

5

Applying control

Time (s)

ndash1

0

1

2

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

(a)

x1

x2

x3

ndash10

0

10

ndash5

0

5

Applying control

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100Time (s)

ndash1

0

1

2

(b)


DSP(TMS320F28335)



PB


XY

Z


8 Complexity


k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


x2(n) + mK2jminus 11113960 1113961



K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)


Complexity 9



6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018









k2x2(n)

K21 ex2(n) minusfk2

k1k3x1(n)x3(n)

K31 minusc

k23x33(n) +

dk3

k1k2x1(n)x2(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


x2(n) + mK2jminus 11113960 1113961



K3j minusc

k23

x3(n) + mK3jminus 11113960 11139613

+dk3


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

(a) (b)

(c) (d)

(e)


Complexity 9



6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018










6 Conclusion


Data Availability




Acknowledgments


References



















10 Complexity










Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018

















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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