research article -...

Research ArticleAbundant Coexisting Multiple Attractorsrsquo Behaviors inThree-Dimensional Sine Chaotic System

Huagan Wu 1 Han Bao 2 Quan Xu 1 and Mo Chen 1

1School of Information Science and Engineering Changzhou University Changzhou 213164 China2College of Automation Engineering Nanjing University of Aeronautics and Astronautics Nanjing 211106 China

Correspondence should be addressed to Mo Chen mchencczueducn

Received 15 October 2019 Revised 2 November 2019 Accepted 15 November 2019 Published 6 December 2019

Guest Editor Viet-anh Pham

Copyright copy 2019 Huagan Wu 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

is paper presents a novel and simple three-dimensional (3-D) chaotic system by introducing two sine nonlinearities into asimple 3-D linear dynamical systeme presented sine system possesses nine equilibrium points consisting of ve index-2 saddlefoci and four index-1 saddle foci which allow the coexistence of various types of disconnected attractors also known asmultistability e coexisting multiple attractors are depicted by the phase plots and attraction basins Coexisting bifurcationmodes triggered by dierent initial values are numerically simulated by two-dimensional bifurcation and complexity plots undertwo sets of initial values and one-dimensional bifurcation plots under three sets of initial values which demonstrate that theabundant coexisting multiple attractorsrsquo behaviors in the presented sine system are related not only to the system parameters butalso to the initial values A simulation-oriented circuit model is synthesized and PSIM (power simulation) screen captures wellvalidate the numerical simulations

1 Introduction

Recently numerous nonlinear dynamical systems have beenreported that they present the coexistence of two or moredisconnected attractors with their isolated attraction basinse coexisting phenomena of two or more attractors havebeen encountered in nonlinear oscillating circuits [1ndash5]biological neuron models [6 7] Hopeld neural networks[8ndash11] vibroimpact system [12] superconducting quantuminterference device oscillators [13] pure mathematicalsystems [14ndash17] and so on is striking phenomenon alsoknown as multistability demonstrates that the system initialvalues do play an important role in the emergence ofcomplex coexisting attractorsrsquo behaviors [18 19] For amultistable dynamical system it is usually challenging topredict the nal steady state to which the dynamical systemwill tend for a given initial value since a small disturbance inthe initial value can alter the steady state of such dynamicalsystems [20ndash23] Multistability has great application po-tentials in the chaos-based secure communication and in-formation encryption [24ndash27] but eiexclcient prediction and

control methods should be employed to make these dy-namical systems in the desired oscillating modes [28ndash32]

Usually an eective method for implementing the initial-related multistability is to lead one two or more generic orextended memristors in various existing circuits and systems[15ndash17 33ndash36] Memristor-based circuits and systems withdierent types of equilibrium points are easy to exhibitcoexisting attractorsrsquo behaviors of multistability Compara-tively speaking another benecial and simple method forgenerating initial value oset-boosted coexisting attractors isto put periodic trigonometric functions into specic oset-boostable dynamical systems [37ndash41] When the cyclic periodsfor the periodic functions are identical any attractor will becopied by periodic oset boosting the initial values [37]However due to the reported oset-boostable dynamicalsystems with self-contained nonlinearities the newly con-structing multistable dynamical systems become relativelycomplicated [38ndash41] not convenient for theoretical analysesand hardware circuit implementations e algebraic sim-plicity of systemrsquos structure and topological complexity ofchaotic attractors are benets for developing chaos-based

HindawiComplexityVolume 2019 Article ID 3687635 11 pageshttpsdoiorg10115520193687635

cryptosystems [42] In this paper based on a simple 3-D lineardynamical system and two newly introduced sine non-linearities a novel and extremely simple 3-D sine chaoticsystem is readily constructed from which abundant coexistingmultiple attractorsrsquo behaviors are observed [43]

ampe rest is organized as follows In Section 2 a novel andsimple 3-D sine chaotic system is presented It has nineequilibrium points consisting of five index-2 saddle foci andfour index-1 saddle foci resulting in the coexistence of up tosix types of disconnected attractors In Section 3 by two-dimensional bifurcation and complexity plots under two setsof initial values and one-dimensional bifurcation plots underthree sets of initial values coexisting bifurcation modes arenumerically simulated to demonstrate the abundant coex-isting multiple attractorsrsquo behaviors In Section 4 with thesimulation-oriented circuit model PSIM screen capturesvalidate the numerical simulations ampe conclusion issummarized in Section 5

2 System Model and Its CoexistingMultiple Attractors

By introducing two sine nonlinearities with two couplingcoefficients into a simple 3-D linear dynamical system anovel 3-D sine chaotic system with simple algebraic equa-tions is easily achieved which is modeled by

_x y + z minus k1 sin(y)

_y minus x + z

_z minus x minus z + k2 sin(x)

(1)

where x y and z are the three state variables and k1 and k2are the two positive constants

ampe presented sine system in (1) is symmetric about theorigin and dissipative ampe symmetric property can bedemonstrated by the invariance of system (1) with respect tothe transformation (x y z)⟷ (minus x minus y minus z) ampe dis-sipativity is explained by

nablaV z _x

zx+

z _y

zy+

z _z

zz minus 1lt 0 (2)

ampus the orbits are finally confined to a specific subsetwith zero volume and its asymptotic motion settles onto astandalone attractor

ampe equilibrium points of the presented sine system in(1) are obtained by solving the following equations

0 y + z minus k1 sin(y)

0 minus x + z

0 minus x minus z + k2 sin(x)

(3)

which is expressed as

E (δ σ δ) (4)

ampe values δ and σ can be yielded by solving the fol-lowing transcendental functions

h1 2δ minus k2 sin(δ) 0 (5)

h2 σ + δ minus k1 sin(σ) 0 (6)

respectivelyampe Jacobian matrix J at E is given as

J

0 1 minus k1 cos(σ) 1

minus 1 0 1

minus 1 + k2 cos(δ) 0 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (7)

so that the characteristic polynomial is derived as

P(λ) λ3 + c1λ2

+ c2λ + c3 0 (8)

where

c1 1

c2 2 minus k1 cos(σ) minus k2 cos(δ)

c3 1 minus k1 cos(σ)1113858 1113859 2 minus k2 cos(δ)1113858 1113859

(9)

ampe above characteristic polynomial implies that Jaco-bian matrix (7) has three nonzero roots For these rootsRouthndashHurwitz conditions are given as

c1 gt 0

c3 gt 0

c1c2 minus c3 gt 0

(10)

ie

1 minus k1 cos(σ)1113858 1113859 2 minus k2 cos(δ)1113858 1113859gt 0

k1 cos(σ) 1 minus k2 cos(δ)1113858 1113859gt 0(11)

If the conditions in (11) are satisfied ie k1 cos(σ)lt 1and k2 cos(δ)lt 1 E is stable leading to the existence of thepoint attractor Otherwise if any one of the conditions in(11) is not satisfied ie k1 cos(σ)lt 1 or k2 cos(δ)lt 1 E isunstable resulting in that unstable behaviors may be trig-gered in the presented sine system

Denote k1 k2 k and take k 36 and 5 as two examplesampe values δ and σ of the equilibrium point E in (4) are theintersection points of two function curves h1 and h2 de-scribed by (5) and (6) as shown in Figures 1(a) and 1(b)respectively from which nine pairs of δ and σ are obtainedby inspecting the intersection points indicating the exis-tence of nine equilibrium points in the presented sinesystem With these equilibrium points the three nonzeroeigenvalues are calculated from the Jacobian matrix J in (7)and the corresponding stabilities can be determined as listedin Table 1 ampe calculation results illustrate that five index-2saddle foci (Index-2 USF for short) and four index-1 saddlefoci (Index-1 USF for short) can be found which couldemerge five disconnected attracting regions when theseattracting regions cannot be linked with each other leadingto the coexistence of disconnected attractors

For k 36 5 and several sets of the initial values (labeledin Figure 2) the phase plots of coexisting multiple attractorsprojected on the x-y plane are depicted in Figures 2(a) and2(b) respectively In Figure 2(a) a chaotic attractor coexistedwith twin small-size period-1 limit cycles twin large-size

2 Complexity

period-1 limit cycles and a relatively larger size period-1 limitcycle By contrast in Figure 2(b) two chaotic attractorscoexisted with twin large-size period-1 limit cycles and arelatively larger size period-1 limit cycle erefore up to six

types of coexisting multiple attractors are numerically dis-closed in the 3-D sine chaotic system because of the attractingand repelling interactions between the ve index-2 saddle fociand four index-1 saddle foci

ndash3 3ndash4

ndash2

0

2

4

P0

P1P3

P6

h1

h2

P2

P4

P5

P7

P8

Index-2 USFIndex-1 USF

δ

σ

ndash2 ndash1 0 1 2

(a)

ndash4 4ndash4

ndash2

0

2

4


P0

P1

P3

P6

h1

h2

P2

P4

P5

P7

P8

δ

σ

ndash2 0 2

(b)

Figure 1 Values δ and σ of the equilibrium points by inspecting the intersections of two function curves h1 and h2 described by (5) and (6)(a) k 36 (b) k 5

Table 1 Equilibrium points eigenvalues and stabilities for k 36 and 5

k Equilibrium points Eigenvalues Stabilities

36

P0 (0 0 0) 10528plusmn j04807 ndash31057 Index-2 USFP12 (0 plusmn24074 0) 13656 ndash11828plusmn j17041 Index-1 USF

P38 (plusmn17659 plusmn17668 plusmn17659) 01205plusmn j19193 ndash12410 Index-2 USFP47 (plusmn17659 plusmn07858 plusmn17659) 13103 ndash11551plusmn j13587 Index-1 USFP56 (plusmn17659 plusmn28378 plusmn17659) 03285plusmn j26672 ndash16569 Index-2 USF

5



x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)

(10minus6 6 0)

(10minus6 minus6 0)

(10minus6 16 0)

y

15

15

10

10

5

5

0

0

ndash5

ndash5

ndash10

ndash10ndash15

ndash15

(a)

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 14 0)

(10minus6 1 0)

(10minus6 minus1 0)

y

15

10

5

0

ndash5

ndash10

ndash15

x151050ndash5ndash10ndash15

(b)

Figure 2 Phase plots of coexisting multiple attractors projected on the x-y plane for dierent initial values (a) For k 36 six types ofcoexisting multiple attractors (b) For k 5 ve types of coexisting multiple attractors

Complexity 3

To detect the attracting regions of the coexisting multipleattractors given in Figure 2 the attraction basins in the initialvalue plane are used to classify different dynamical behaviors[44] For the two coupling coefficients used in Figure 2 theattraction basins in the x(0)-y(0) plane with z(0) 10minus 6 aredepicted in Figure 3 ampe attracting regions painted bydifferent colors represent the initial value regions corre-sponding to different long-term oscillating states iecoexisting multistable states which are identical with thecolored trajectories appearing in Figure 2 ampus the at-traction basins show the relatively complicated manifoldstructures along with basin boundaries Meanwhile thenumerical results in Figure 3 demonstrate the emergence ofmultistability in the presented sine system

It can be concluded that due to the appearance of fiveindex-2 saddle foci and four index-1 saddle foci and theirinteractions with each other some disconnected attractingregions are thereby formed in the neighborhoods aroundthese unstable saddle foci resulting in the generation ofcoexisting multiple disconnected attractors

3 Initial Values-Related Coexisting MultipleBifurcation Modes

Because the presented sine system in (1) is symmetric aboutthe origin the disconnected attracting regions have locallysymmetric behaviors which are well exhibited in Figures 2and 3 For this reason and convenient analysis three sets ofthe initial values (10minus 6 0 0) (10minus 6 3 0) and (10minus 6 ndash3 0) areconsidered in next numerical plots ampe fourth-orderRungendashKutta algorithm with the 001 s time step and (700 s800 s) time interval is used for depicting the phase plots andbifurcation plots whereas the fourth-order RungendashKutta-basedWolfrsquos method with the 001 s time step and 20 ks timeend is adopted for calculating the Lyapunov exponents

Firstly two-dimensional bifurcation plots (bifurcationdiagrams and dynamical maps) [23] are employed to showcomplex dynamical behaviors in the presented sine systemintuitively as shown in Figures 4 and 5 Here both thecoupling coefficients k1 and k2 are simultaneously increasedin the region [2 8] and two sets of initial values (10minus 6 0 0)and (10minus 6 3 0) are chosen Note that the exhibited two-dimensional bifurcation behaviors are similar to each otherfor the initial values (10minus 6 3 0) and (10minus 6 ndash3 0) due to thesystem symmetry

As shown in Figure 4 the two-dimensional bifurcationdiagrams in the k1-k2 parameter plane are obtained bycalculating the periodicities of state variable x whichdemonstrate rich and complex coexisting dynamical be-haviors related to system parameters and initial values ampestable points and chaotic attractors distribute in the black-and red-colored regions respectively and the periodicattractors with different periodicities situate in the othercolored regions Comparing Figure 4(b) with Figure 4(a)there is a big difference between the two dynamical be-haviors in the lower right regions which is triggered by theinitial values-dependent multistability in the presented sinesystem leading to the coexistence of multiple bifurcationmodes

As shown in Figure 5 the two-dimensional dynamicalmaps in the k1-k2 parameter plane under two sets of initialvalues are depicted by evaluating the values of the largestLyapunov exponent ampe yellow-red-white colored regionswith different positive values of the largest Lyapunov ex-ponent represent different chaotic behaviors the black-colored regions with different negative values of the largestLyapunov exponent only stand for stable point behaviorsand the black-yellow colored regions with the zero largestLyapunov exponent represent different periodic behaviorsIn a similar manner the dynamical behaviors described bythe dynamical maps in Figures 5(a) and 5(b) are of greatdifference which manifest how coexisting dynamical be-haviors evolve for different initial values

Similarly the two-dimensional spectral entropy-basedcomplexity plots in the k1-k2 parameter plane are displayedin Figure 6 where two sets of initial values (10minus 6 0 0) and(10minus 6 3 0) are employed On the basis of the Fouriertransform [32 45] the complexity values are obtained bycalculating the spectral entropy of the time sequence of thevariable x ampe relatively large complexity value in Figure 6indicates the appearance of an irregularly chaotic sequencewhereas the relatively small complexity value in Figure 6represents the occurrence of a regularly periodic sequenceFor the two sets of different initial values there are somedifferences in the complexity plots between Figures 6(a) and6(b) implying that the system initial values have great effectson the dynamical behaviors of the presented sine system

amperefore the dynamical maps shown in Figure 5 andcomplexity plots shown in Figure 6 can reflect the dynamicalevolutions with the variations of the system parameters andinitial values which are the effective supplements to confirmthe coexisting dynamical behaviors depicted by the bi-furcation diagrams in Figure 4

To visualize the coexisting multiple bifurcation modesrelated to the initial values three sets of initial values (10minus 6 00) (10minus 6 3 0) and (10minus 6 ndash3 0) are considered and both thecoupling coefficients k1 and k2 are simultaneously increasedin the region [2 8] Denote k1 k2 k as a bifurcation pa-rameter ampe one-dimensional bifurcation plots with thevariation of the system parameter k are shown in Figure 7 InFigure 7(a) the bifurcation diagrams drawn by the blackblue and red trajectories correspond to those initiated fromthe initial values (10minus 6 0 0) (10minus 6 3 0) and (10minus 6 ndash3 0)respectively And in Figure 7(b) the first two Lyapunovexponents associated with three sets of initial values aredrafted in the upper middle and bottom of Figure 7(b)which entirely match with the bifurcation diagrams inFigure 7(a) amperefore when more initial values are con-sidered more complicated coexisting multiple bifurcationmodes can be revealed in the presented sine system

Observed from Figure 7 abundant coexisting multipleattractorsrsquo behaviors related to the initial values areexhibited including stable points periodic oscillations andchaotic oscillations along with period-doubling bifurcationstangent bifurcations and crisis scenarios When two sets ofinitial values (10minus 6 3 0) and (10minus 6 ndash3 0) are chosen both thedepicted dynamical behaviors in Figure 7 are basicallyidentical over the entire parameter region with only slight

4 Complexity

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(b)

Figure 4 Two-dimensional bifurcation diagrams in the k1-k2 parameter plane through calculation of the periodicities of the state variable xunder two sets of initial values (a) Initial values (10minus 6 0 0) (b) Initial values (10minus 6 3 0)

ndash12ndash12

ndash8

ndash4

0

4

8

12

ndash8 ndash4 0 4 8 12x (0)

y (0)

(a)

ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(b)

Figure 3 Two attraction basins in the x(0)-y(0) plane with z(0) 10minus 6 and the painted colors correspond to the colored motion orbitsshown in Figure 2 (a) Attraction basin for k 36 (b) Attraction basin for k 5

22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(b)

Figure 5 Two-dimensional dynamical maps in the k1-k2 parameter plane by evaluating the values of the largest Lyapunov exponent undertwo sets of initial values (a) Initial values (10minus 6 0 0) (b) Initial values (10minus 6 3 0)

Complexity 5

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(b)

Figure 6 Two-dimensional spectral entropy-based complexity plots for the variable x sequence in the k1-k2 parameter plane under two setsof initial values (a) Initial values (10minus 6 0 0) (b) Initial values (10minus 6 3 0)

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

k

y max

18

15

9

3

ndash3

ndash92 3 4 5 6 7 8

(a)

k

ndash05

0

05(10minus6 minus3 0)

ndash05

0

05(10minus6 3 0)

Lyap

unov

expo

nent

s

ndash05

0

05(10minus6 0 0)

2 3 4 5 6 7 8

(b)

Figure 7 For three sets of initial values (10minus 6 0 0) (10minus 6 3 0) and (10minus 6 ndash3 0) one-dimensional bifurcation plots with the variation of thesystem parameter k (a) Bifurcation diagrams of the maxima ymax of the variable y (b) First two Lyapunov exponents

y

x

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

6

6

4

4

2

2

0

0

ndash2

ndash2

ndash4

ndash4ndash6

ndash6

(a)

y

x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)9

6

3

0

ndash3

ndash6

ndash99630ndash3ndash6ndash9

(b)

Figure 8 Continued

6 Complexity

dierences in the parameter region (694 758) Howeverwhen the other two sets of initial values (10minus 6 0 0) and (10minus 6 30) are chosen both the depicted dynamical behaviors inFigure 7 have big dierences in the parameter region (296524) As the parameter k is increased in this parameter regionthe moving orbit for (10minus 6 0 0) goes into chaotic oscillatingstate at k 301 via period-doubling bifurcation route andmutates into periodic oscillating state at k 376 via chaoscrisis whereas the moving orbit for (10minus 6 3 0) turns intoperiodic oscillating state from stable resting state at k 314and enters into chaotic oscillating state at k 434 via period-doubling bifurcation route with two relatively larger periodicwindows Of course in the parameter region (694 758) someslight dierences between the depicted dynamical behaviorsunder two sets of initial values (10minus 6 0 0) and (10minus 6 3 0) canbe seen for the presented sine system as well

Except for the two examples in Figure 2 other examplesto exhibit coexisting multiple attractorsrsquo behaviors are givenin Figure 8 where four sets of phase plots in the x-y plane areprovided together for the initial values (10minus 6 0 0) (10minus 6 30) and (10minus 6 ndash3 0) When k 3 the coexistence of a period-4 limit cycle and a pair of symmetric points is exhibited inFigure 8(a) When k 45 the coexistence of a large sizeperiod-1 limit cycle and a pair of symmetric period-3 limitcycles is demonstrated in Figure 8(b) When k 7 the co-existence of a chaotic attractor and a period-5 limit cycle isillustrated in Figure 8(c) However when k 8 the co-existence of two chaotic attractors with dierent topologiesis disclosed in Figure 8(d) Consequently various types ofcoexisting attractorsrsquo behaviors can be found in the pre-sented sine system

4 Validations by the Simulation-OrientedCircuit Model

By employing PSIM Version 903 software the simulation-oriented circuit model for implementation of the presentedsine system is synthesized and its screen shot is given inFigure 9 in which three operation channels containing three

integrators three inverters and two sine function convertersare used to implement three state variables x y and zrespectively

Based on the simulation-oriented circuit model shown inFigure 9 the state equations for the capacitor voltages vx vyand vz are described by

RCdvxdt

vy + vz minusR

Rk1sin vy( )

RCdvydt

minus vx + vz

RCdvzdt

minus vx minus vz minusR

Rk2sin vx( )

(12)

where C1C2C3C Rk1Rk1 and Rk2Rk2 WhenRC 10 kΩtimes 10 nF 100 μs ie R 10 kΩ and C 10 nFthe circuit parameters Rk1 and Rk2 for PSIM circuit simu-lations can be conveniently determined

According to the system parameters k1 and k2 and theinitial values used in Figure 2 the circuit parameters Rk1 andRk2 have the same values ie Rk1Rk2 WhenRk1Rk2 278 kΩ and 2 kΩ respectively PSIM screencaptures are obtained in Figure 10 where the initial voltagesvx(0) and vz(0) of the capacitors C1 and C3 are always xedas 1 μV and 0V respectively and only the initial voltagevy(0) of the capacitor C2 is adjusted as dierent initialvalues

Similarly based on the system parameters k1 and k2 andthree sets of initial values used in Figure 8 the circuit pa-rameters are selected as Rk1Rk2 333 kΩ 222 kΩ 143 kΩand 125 kΩ respectively e corresponding PSIM screencaptures are attached in Figure 11 where the initial voltagesvx(0) vy(0) and vz(0) of the capacitors C1 C2 and C3 areassigned as vx(0) 1μV vy(0) 3V (or 0V and ndash3V) andvz(0) 0V respectively

PSIM circuit simulations in Figure 11(d) are slightlydierent from MATLAB numerical simulations inFigure 8(d) which are mainly caused by the inconsistently

y

x

(10minus6 minus3 0) (10minus6 3 0)

(10minus6 0 0)10

10

5

5

0

0

ndash5

ndash5ndash10

ndash10

(c)

y

x


(10minus6 0 0)

18

18

12

12

6

6

0

0

ndash6

ndash6

ndash12

ndash12ndash18

ndash18

(d)

Figure 8 Phase plots of coexisting attractors in the x-y plane for dierent values of the parameter k (a) Period-4 limit cycle coexistedwith a pair of symmetric points at k 3 (b) Large-size period-1 limit cycle coexisted with a pair of symmetric period-3 limit cycles atk 45 (c) Chaotic attractor coexisted with period-5 limit cycle at k 7 (d) Coexisting chaotic attractors with two topologies at k 8

Complexity 7

transient behaviors due to the existence of simulation errors[46] Ignoring the tiny dierences between MATLAB nu-merical simulations and PSIM circuit simulations the re-sults in Figures 10 and 11 eectively validate the coexistingattractorsrsquo behaviors disclosed in Figures 2 and 8

Besides it should be mentioned that the sine functionterms are the two key units for realizing the proposed 3-Dsine chaotic system In the analog circuit experiments [47]the sine function terms can be physically implemented using

two AD639AD trigonometric function converters But thesystem initials corresponding to the initial capacitor volt-ages are hardly set in the experimental measurements Incontrast in the digital circuit experiments [48] the sinefunction terms can be directly achieved by calling IP cores inCORDIC library of FPGA and the system initials can bereadily preset erefore a feasible way to realize the pro-posed 3-D sine chaotic system could be implemented on theFPGA which is addressed in our future paper

C1

C2

C3Rk2

10n

ndash+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

+

10n

10n

10k

sin(x)m

sin(x)m

10k

10k 10k10k

10k

10k

10k

10k

Vy

Vz

10k

10k

U2

U4

U6U5

VxVV

V

10k

Rk1U1

U3

deg

deg

deg

deg deg

deg

deg

deg

deg

deg deg

deg

deg

deg

deg

degdeg

Figure 9 Screen shot of PSIM simulation-oriented circuit model for implementation of the presented sine system

ndash15ndash15

75

0

15

75

v y

vx

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 0V 0V)

(1μV 6V 0V)

(1μV minus6V 0V)

(1μV 16V 0V)

ndash10 ndash5 0 5 10 15

(a)

ndash15ndash15

75

0

15

75

v y

vxndash10 ndash5 0 5 10 15

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 1V 0V)

(1μV minus1V 0V)

(1μV 14V 0V)

(b)

Figure 10 PSIM screen captures of coexisting multiple attractors in the vx-vy plane for dierent initial values (a) For Rk1Rk2 278 kΩsix types of coexisting multiple attractors (b) For Rk1Rk2 2 kΩ ve types of coexisting multiple attractors

8 Complexity

5 Conclusion

e autonomous chaotic systems can generate the con-ventional self-excited attractors as their oscillations areexcited from the unstable determined equilibrium pointse mechanism for constructing chaotic systems withcoexisting multiple attractors is based on the fact that thesystem equilibrium points can be reinstalled by newly in-troduced sine nonlinearities leading to the great variationsof their number characteristics and distributions [49]erefore by introducing two sine nonlinearities into asimple 3-D linear dynamical system this paper presented anovel and simple 3-D sine chaotic systemwith the reinstalledve index-2 saddle foci and four index-1 saddle foci fromwhich the abundant coexisting multiple attractorsrsquo behaviorswere thereby revealed by numerical simulations such asphase plots attraction basins two-dimensional bifurcationand complexity plots and one-dimensional bifurcationplots and nally validated by PSIM circuit simulations e

algebraic simplicity of system structure and topologicalcomplexity of chaotic attractor are a long-term goal forseeking a new chaotic system with coexisting behaviorswhich could acquire wide interest for its chaos-based en-gineering applications [42 50]

Data Availability

e data used to support the ndings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conumlicts of interest

Acknowledgments

is research was supported by the grants from the NationalNatural Science Foundations of China under Grant nos

6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4

ndash6ndash6 ndash4 ndash2 0

vx

v y

2 4 6

(a)

9

45

ndash45

ndash9ndash9 ndash6 ndash3 0 3 6 9

0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5

ndash10ndash10 ndash5 0 5 10

(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9

ndash18ndash18 ndash12 12 18ndash6 0 6

0

(1μV 0V 0V)

(1μV 3V 0V)(1μV ndash3V 0V)

vx

v y

(d)

Figure 11 PSIM screen captures of coexisting attractors in the vx minus vx plane for dierent values of Rk1 and Rk2 (a) Period-4 limit cyclecoexisted with a pair of symmetric points at Rk1Rk2 333 kΩ (b) Large-size period-1 limit cycle coexisted with a pair of symmetric period-3 limit cycles at Rk1Rk2 222 kΩ (c) Chaotic attractor coexisted with period-5 limit cycle at Rk1Rk2143 kΩ (d) Coexisting chaoticattractors with two topologies at Rk1Rk2125 kΩ

Complexity 9

51607013 61601062 and 61801054 and Natural ScienceFoundation of Jiangsu Province China under Grant noBK20191451

References

[1] G H Kom J Kengne J R Mboupda Pone G Kenne andA B Tiedeu ldquoAsymmetric double strange attractors in asimple autonomous jerk circuitrdquo Complexity vol 2018 Ar-ticle ID 4658785 16 pages 2018

[2] L Zhou C H Wang X Zhang and W Yao ldquoVariousattractors coexisting attractors and antimonotonicity in asimple fourth-order memristive Twin-T oscillatorrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 4Article ID 1850050 2018

[3] M Chen Q Xu Y Lin and B Bao ldquoMultistability induced bytwo symmetric stable node-foci in modified canonical Chuarsquoscircuitrdquo Nonlinear Dynamics vol 87 no 2 pp 789ndash8022017

[4] A T Azar N M Adele T Alain R Kengne andF H Bertrand ldquoMultistability analysis and function pro-jective synchronization in relay coupled oscillatorsrdquo Com-plexity vol 2018 Article ID 3286070 12 pages 2018

[5] N Stankevich and E Volkov ldquoMultistability in a three-di-mensional oscillator tori resonant cycles and chaosrdquo Non-linear Dynamics vol 94 no 4 pp 2455ndash2467 2018

[6] B C Bao A H Hu H Bao Q Xu M Chen and H G Wuldquoampree-dimensional memristive Hindmarsh-Rose neuronmodel with hidden coexisting asymmetric behaviorsrdquo Com-plexity vol 2018 Article ID 3872573 11 pages 2018

[7] H Bao W Liu and A Hu ldquoCoexisting multiple firingpatterns in two adjacent neurons coupled by memristiveelectromagnetic inductionrdquo Nonlinear Dynamics vol 95no 1 pp 43ndash56 2019

[8] Z T Njitacke and J Kengne ldquoComplex dynamics of a 4DHopfield neural networks (HNNs) with a nonlinear synapticweight coexistence of multiple attractors and remergingFeigenbaum treesrdquo AEUmdashInternational Journal of Electronicsand Communications vol 93 pp 242ndash252 2018

[9] B C Bao H Qian Q Xu M Chen J Wang and Y J YuldquoCoexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural networkrdquo Frontiers inComputational Neuroscience vol 11 no 81 pp 1ndash14 2017

[10] K Rajagopal J M Munoz-Pacheco V-T PhamD V Hoang F E Alsaadi and F E Alsaadi ldquoA Hopfieldneural network with multiple attractors and its FPGA designrdquo1eEuropean Physical Journal Special Topics vol 227 no 7ndash9pp 811ndash820 2018

[11] C Chen J Chen H Bao M Chen and B Bao ldquoCoexistingmulti-stable patterns in memristor synapse-coupled Hopfieldneural network with two neuronsrdquo Nonlinear Dynamicsvol 95 no 4 pp 3385ndash3399 2019

[12] Y Zhang and G Luo ldquoMultistability of a three-degree-of-freedom vibro-impact systemrdquo Communications in NonlinearScience and Numerical Simulation vol 57 pp 331ndash341 2018

[13] J Hizanidis N Lazarides and G P Tsironis ldquoFlux bias-controlled chaos and extreme multistability in SQUID os-cillatorsrdquo Chaos An Interdisciplinary Journal of NonlinearScience vol 28 no 6 Article ID 063117 2018

[14] Q Lai P D K Kuate F Liu and H H C Iu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs 2019

[15] M Chen Y Feng H Bao et al ldquoState variable mappingmethod for studying initial-dependent dynamics in mem-ristive hyper-jerk system with line equilibriumrdquo Chaos Sol-itons amp Fractals vol 115 pp 313ndash324 2018

[16] Z T Njitacke J Kengne R W Tapche and F B PelapldquoUncertain destination dynamics of a novel memristive 4Dautonomous systemrdquo Chaos Solitons amp Fractals vol 107pp 177ndash185 2018

[17] H Bao N Wang B Bao M Chen P Jin and G WangldquoInitial condition-dependent dynamics and transient periodin memristor-based hypogenetic jerk system with four lineequilibriardquo Communications in Nonlinear Science and Nu-merical Simulation vol 57 pp 264ndash275 2018

[18] A N Pisarchik and U Feudel ldquoControl of multistabilityrdquoPhysics Reports vol 540 no 4 pp 167ndash218 2014

[19] P R Sharma M D Shrimali A Prasad N V Kuznetsov andG A Leonov ldquoControl of multistability in hidden attractorsrdquo1e European Physical Journal Special Topics vol 224 no 8pp 1485ndash1491 2015

[20] M Chen M Sun B Bao H Wu Q Xu and J WangldquoControlling extreme multistability of memristor emulator-based dynamical circuit in flux-charge domainrdquo NonlinearDynamics vol 91 no 2 pp 1395ndash1412 2018

[21] F Hegedus W Lauterborn U Parlitz and R Mettin ldquoNon-feedback technique to directly control multistability innonlinear oscillators by dual-frequency drivingrdquo NonlinearDynamics vol 94 no 1 pp 273ndash293 2018

[22] K Yadav A Prasad and M D Shrimali ldquoControl of coex-isting attractors via temporal feedbackrdquo Physics Letters Avol 382 no 32 pp 2127ndash2132 2018

[23] M Chen M X Sun H Bao Y H Hu and B C Bao ldquoFlux-charge analysis of two-memristor-based Chuarsquos circuit di-mensionality decreasing model for detecting extreme multi-stabilityrdquo IEEE Transactions on Industrial Electronics vol 67no 3 pp 2197ndash2206 2019

[24] Z Wang A Akgul V-T Pham and S Jafari ldquoChaos-basedapplication of a novel no-equilibrium chaotic system withcoexisting attractorsrdquo Nonlinear Dynamics vol 89 no 3pp 1877ndash1887 2017

[25] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended Lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[26] G Peng and F Min ldquoMultistability analysis circuit imple-mentations and application in image encryption of a novelmemristive chaotic circuitrdquo Nonlinear Dynamics vol 90no 3 pp 1607ndash1625 2017

[27] C Li F H Min Q S Jin and H Y Ma ldquoExtreme multi-stability analysis of memristor-based chaotic system and itsapplication in image decryptionrdquo AIP Advances vol 7 no 12Article ID 125204 2017

[28] F Yuan G Y Wang and X W Wang ldquoChaotic oscillatorcontaining memcapacitor and meminductor and its di-mensionality reduction analysisrdquo Chaos An InterdisciplinaryJournal of Nonlinear Science vol 27 no 3 Article ID 0331032017

[29] M Chen Y Feng H Bao B C Bao H G Wu and Q XuldquoHybrid state variable incremental integral for reconstructingextreme multistability in memristive jerk system with cubicnonlinearityrdquo Complexity vol 2019 Article ID 854947216 pages 2019

[30] H Bao T Jiang K B Chu M Chen Q Xu and B C BaoldquoMemristor-based canonical Chuarsquos circuit extreme multi-stability in voltage-current domain and its controllability in

10 Complexity

flux-charge domainrdquo Complexity vol 2018 Article ID5935637 13 pages 2018

[31] M Chen B C Bao T Jiang et al ldquoFlux-Charge analysis ofinitial state-dependent dynamical behaviors of a memristoremulator-based chuarsquos circuitrdquo International Journal of Bi-furcation and Chaos vol 28 no 10 Article ID 1850120 2018

[32] H Bao W Liu and M Chen ldquoHidden extreme multistabilityand dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh-Nagumo circuitrdquo Non-linear Dynamics vol 96 no 3 pp 1879ndash1894 2019

[33] Q Xu Y Lin B Bao and M Chen ldquoMultiple attractors in anon-ideal active voltage-controlled memristor based Chuarsquoscircuitrdquo Chaos Solitons amp Fractals vol 83 pp 186ndash200 2016

[34] B Bao T Jiang G Wang P Jin H Bao and M Chen ldquoTwo-memristor-based Chuarsquos hyperchaotic circuit with planeequilibrium and its extreme multistabilityrdquo Nonlinear Dy-namics vol 89 no 2 pp 1157ndash1171 2017

[35] L Wang S Zhang Y-C Zeng and Z-J Li ldquoGeneratinghidden extreme multistability in memristive chaotic oscillatorvia micro-perturbationrdquo Electronics Letters vol 54 no 13pp 808ndash810 2018

[36] J Kengne Z T Njitacke and H B Fotsin ldquoDynamicalanalysis of a simple autonomous jerk system with multipleattractorsrdquo Nonlinear Dynamics vol 83 no 1-2 pp 751ndash7652016

[37] C Li and J C Sprott ldquoAn infinite 3-D quasiperiodic lattice ofchaotic attractorsrdquo Physics Letters A vol 382 no 8pp 581ndash587 2018

[38] J Sun X Zhao J Fang and Y Wang ldquoAutonomousmemristor chaotic systems of infinite chaotic attractors andcircuitry realizationrdquo Nonlinear Dynamics vol 94 no 4pp 2879ndash2887 2018

[39] C Li Y Xu G Chen Y Liu and J Zheng ldquoConditionalsymmetry bond for attractor growingrdquo Nonlinear Dynamicsvol 95 no 2 pp 1245ndash1256 2019

[40] Q Lai C Chen X-W Zhao J Kengne and C VolosldquoConstructing chaotic system with multiple coexistingattractorsrdquo IEEE Access vol 7 pp 24051ndash24056 2019

[41] C Li W Joo-Chen ampio J C Sprott H H-C Iu and Y XuldquoConstructing infinitely many attractors in a programmablechaotic circuitrdquo IEEE Access vol 6 pp 29003ndash29012 2018

[42] G Alvarez and S Li ldquoSome basic cryptographic requirementsfor chaos-based cryptosystemsrdquo International Journal of Bi-furcation and Chaos vol 16 no 8 pp 2129ndash2151 2006

[43] T F Fonzin K Srinivasan J Kengne and F B PelapldquoCoexisting bifurcations in a memristive hyperchaotic os-cillatorrdquo AEUmdashInternational Journal of Electronics andCommunications vol 90 pp 110ndash122 2018

[44] C C Strelioff and AW Hubler ldquoMedium-term prediction ofchaosrdquo Physical Review Letters vol 96 no 4 Article ID044101 2006

[45] H Bao M Chen H Wu and B Bao ldquoMemristor initial-boosted coexisting plane bifurcations and its extreme multi-stability reconstitution in two-memristor-based dynamicalsystemrdquo Science China Technological Sciences 2019

[46] N V Kuznetsov G A Leonov M V Yuldashev andR V Yuldashev ldquoHidden attractors in dynamical models ofphase-locked loop circuits limitations of simulation inMATLAB and SPICErdquo Communications in Nonlinear Scienceand Numerical Simulation vol 51 pp 39ndash49 2017

[47] Q Lai A Akgul C Li G Xu and U Ccedilavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and S-Box designrdquo Entropy vol 20 no 1p 12 2018

[48] B C Bao Q F Yang L Zhu et al ldquoChaotic bursting dy-namics and coexisting multistable firing patterns in 3D au-tonomous MorrisndashLecar model and microcontroller-basedvalidationsrdquo International Journal of Bifurcation and Chaosvol 29 no 10 Article ID 1950134 2019

[49] V T Pham C Volos T Kapitaniak S Jafari and X WangldquoDynamics and circuit of a chaotic system with a curve ofequilibrium pointsrdquo International Journal of Electronicsvol 105 no 3 pp 385ndash397 2018

[50] Z Y Hua Y C Zhou and B C Bao ldquoTwo-dimensional sinechaotification system with hardware implementationrdquo IEEETransactions on Industrial Informatics 2019

Complexity 11

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


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


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

cryptosystems [42] In this paper based on a simple 3-D lineardynamical system and two newly introduced sine non-linearities a novel and extremely simple 3-D sine chaoticsystem is readily constructed from which abundant coexistingmultiple attractorsrsquo behaviors are observed [43]

ampe rest is organized as follows In Section 2 a novel andsimple 3-D sine chaotic system is presented It has nineequilibrium points consisting of five index-2 saddle foci andfour index-1 saddle foci resulting in the coexistence of up tosix types of disconnected attractors In Section 3 by two-dimensional bifurcation and complexity plots under two setsof initial values and one-dimensional bifurcation plots underthree sets of initial values coexisting bifurcation modes arenumerically simulated to demonstrate the abundant coex-isting multiple attractorsrsquo behaviors In Section 4 with thesimulation-oriented circuit model PSIM screen capturesvalidate the numerical simulations ampe conclusion issummarized in Section 5

2 System Model and Its CoexistingMultiple Attractors

By introducing two sine nonlinearities with two couplingcoefficients into a simple 3-D linear dynamical system anovel 3-D sine chaotic system with simple algebraic equa-tions is easily achieved which is modeled by

_x y + z minus k1 sin(y)

_y minus x + z

_z minus x minus z + k2 sin(x)

(1)

where x y and z are the three state variables and k1 and k2are the two positive constants

ampe presented sine system in (1) is symmetric about theorigin and dissipative ampe symmetric property can bedemonstrated by the invariance of system (1) with respect tothe transformation (x y z)⟷ (minus x minus y minus z) ampe dis-sipativity is explained by

nablaV z _x

zx+

z _y

zy+

z _z

zz minus 1lt 0 (2)

ampus the orbits are finally confined to a specific subsetwith zero volume and its asymptotic motion settles onto astandalone attractor

ampe equilibrium points of the presented sine system in(1) are obtained by solving the following equations

0 y + z minus k1 sin(y)

0 minus x + z

0 minus x minus z + k2 sin(x)

(3)

which is expressed as

E (δ σ δ) (4)

ampe values δ and σ can be yielded by solving the fol-lowing transcendental functions

h1 2δ minus k2 sin(δ) 0 (5)

h2 σ + δ minus k1 sin(σ) 0 (6)

respectivelyampe Jacobian matrix J at E is given as

J

0 1 minus k1 cos(σ) 1

minus 1 0 1

minus 1 + k2 cos(δ) 0 minus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (7)

so that the characteristic polynomial is derived as

P(λ) λ3 + c1λ2

+ c2λ + c3 0 (8)

where

c1 1

c2 2 minus k1 cos(σ) minus k2 cos(δ)

c3 1 minus k1 cos(σ)1113858 1113859 2 minus k2 cos(δ)1113858 1113859

(9)

ampe above characteristic polynomial implies that Jaco-bian matrix (7) has three nonzero roots For these rootsRouthndashHurwitz conditions are given as

c1 gt 0

c3 gt 0

c1c2 minus c3 gt 0

(10)

ie

1 minus k1 cos(σ)1113858 1113859 2 minus k2 cos(δ)1113858 1113859gt 0

k1 cos(σ) 1 minus k2 cos(δ)1113858 1113859gt 0(11)

If the conditions in (11) are satisfied ie k1 cos(σ)lt 1and k2 cos(δ)lt 1 E is stable leading to the existence of thepoint attractor Otherwise if any one of the conditions in(11) is not satisfied ie k1 cos(σ)lt 1 or k2 cos(δ)lt 1 E isunstable resulting in that unstable behaviors may be trig-gered in the presented sine system

Denote k1 k2 k and take k 36 and 5 as two examplesampe values δ and σ of the equilibrium point E in (4) are theintersection points of two function curves h1 and h2 de-scribed by (5) and (6) as shown in Figures 1(a) and 1(b)respectively from which nine pairs of δ and σ are obtainedby inspecting the intersection points indicating the exis-tence of nine equilibrium points in the presented sinesystem With these equilibrium points the three nonzeroeigenvalues are calculated from the Jacobian matrix J in (7)and the corresponding stabilities can be determined as listedin Table 1 ampe calculation results illustrate that five index-2saddle foci (Index-2 USF for short) and four index-1 saddlefoci (Index-1 USF for short) can be found which couldemerge five disconnected attracting regions when theseattracting regions cannot be linked with each other leadingto the coexistence of disconnected attractors

For k 36 5 and several sets of the initial values (labeledin Figure 2) the phase plots of coexisting multiple attractorsprojected on the x-y plane are depicted in Figures 2(a) and2(b) respectively In Figure 2(a) a chaotic attractor coexistedwith twin small-size period-1 limit cycles twin large-size

2 Complexity



ndash3 3ndash4

ndash2

0

2

4

P0

P1P3

P6

h1

h2

P2

P4

P5

P7

P8


δ

σ

ndash2 ndash1 0 1 2

(a)

ndash4 4ndash4

ndash2

0

2

4


P0

P1

P3

P6

h1

h2

P2

P4

P5

P7

P8

δ

σ

ndash2 0 2

(b)




36



5



x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)

(10minus6 6 0)

(10minus6 minus6 0)

(10minus6 16 0)

y

15

15

10

10

5

5

0

0

ndash5

ndash5

ndash10

ndash10ndash15

ndash15

(a)

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 14 0)

(10minus6 1 0)

(10minus6 minus1 0)

y

15

10

5

0

ndash5

ndash10

ndash15


(b)


Complexity 3












4 Complexity

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(b)


ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(a)

ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(b)


22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(b)


Complexity 5

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(b)


(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

k

y max

18

15

9

3

ndash3

ndash92 3 4 5 6 7 8

(a)

k

ndash05

0


ndash05

0

05(10minus6 3 0)

Lyap

unov

expo

nent

s

ndash05

0

05(10minus6 0 0)

2 3 4 5 6 7 8

(b)


y

x

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

6

6

4

4

2

2

0

0

ndash2

ndash2

ndash4

ndash4ndash6

ndash6

(a)

y

x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)9

6

3

0

ndash3

ndash6


(b)

Figure 8 Continued

6 Complexity







RCdvxdt

vy + vz minusR

Rk1sin vy( )

RCdvydt

minus vx + vz

RCdvzdt


Rk2sin vx( )

(12)





y

x


(10minus6 0 0)10

10

5

5

0

0

ndash5

ndash5ndash10

ndash10

(c)

y

x


(10minus6 0 0)

18

18

12

12

6

6

0

0

ndash6

ndash6

ndash12

ndash12ndash18

ndash18

(d)


Complexity 7




C1

C2

C3Rk2

10n

ndash+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

+

10n

10n

10k

sin(x)m

sin(x)m

10k

10k 10k10k

10k

10k

10k

10k

Vy

Vz

10k

10k

U2

U4

U6U5

VxVV

V

10k

Rk1U1

U3

deg

deg

deg

deg deg

deg

deg

deg

deg

deg deg

deg

deg

deg

deg

degdeg


ndash15ndash15

75

0

15

75

v y

vx

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 0V 0V)

(1μV 6V 0V)

(1μV minus6V 0V)

(1μV 16V 0V)


(a)

ndash15ndash15

75

0

15

75

v y


(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 1V 0V)

(1μV minus1V 0V)

(1μV 14V 0V)

(b)


8 Complexity

5 Conclusion



Data Availability




Acknowledgments


6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4


vx

v y

2 4 6

(a)

9

45

ndash45


0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5


(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9


0

(1μV 0V 0V)


vx

v y

(d)


Complexity 9


References































10 Complexity






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018










ndash3 3ndash4

ndash2

0

2

4

P0

P1P3

P6

h1

h2

P2

P4

P5

P7

P8


δ

σ

ndash2 ndash1 0 1 2

(a)

ndash4 4ndash4

ndash2

0

2

4


P0

P1

P3

P6

h1

h2

P2

P4

P5

P7

P8

δ

σ

ndash2 0 2

(b)




36



5



x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)

(10minus6 6 0)

(10minus6 minus6 0)

(10minus6 16 0)

y

15

15

10

10

5

5

0

0

ndash5

ndash5

ndash10

ndash10ndash15

ndash15

(a)

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 14 0)

(10minus6 1 0)

(10minus6 minus1 0)

y

15

10

5

0

ndash5

ndash10

ndash15


(b)


Complexity 3












4 Complexity

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(b)


ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(a)

ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(b)


22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(b)


Complexity 5

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(b)


(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

k

y max

18

15

9

3

ndash3

ndash92 3 4 5 6 7 8

(a)

k

ndash05

0


ndash05

0

05(10minus6 3 0)

Lyap

unov

expo

nent

s

ndash05

0

05(10minus6 0 0)

2 3 4 5 6 7 8

(b)


y

x

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

6

6

4

4

2

2

0

0

ndash2

ndash2

ndash4

ndash4ndash6

ndash6

(a)

y

x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)9

6

3

0

ndash3

ndash6


(b)

Figure 8 Continued

6 Complexity







RCdvxdt

vy + vz minusR

Rk1sin vy( )

RCdvydt

minus vx + vz

RCdvzdt


Rk2sin vx( )

(12)





y

x


(10minus6 0 0)10

10

5

5

0

0

ndash5

ndash5ndash10

ndash10

(c)

y

x


(10minus6 0 0)

18

18

12

12

6

6

0

0

ndash6

ndash6

ndash12

ndash12ndash18

ndash18

(d)


Complexity 7




C1

C2

C3Rk2

10n

ndash+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

+

10n

10n

10k

sin(x)m

sin(x)m

10k

10k 10k10k

10k

10k

10k

10k

Vy

Vz

10k

10k

U2

U4

U6U5

VxVV

V

10k

Rk1U1

U3

deg

deg

deg

deg deg

deg

deg

deg

deg

deg deg

deg

deg

deg

deg

degdeg


ndash15ndash15

75

0

15

75

v y

vx

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 0V 0V)

(1μV 6V 0V)

(1μV minus6V 0V)

(1μV 16V 0V)


(a)

ndash15ndash15

75

0

15

75

v y


(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 1V 0V)

(1μV minus1V 0V)

(1μV 14V 0V)

(b)


8 Complexity

5 Conclusion



Data Availability




Acknowledgments


6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4


vx

v y

2 4 6

(a)

9

45

ndash45


0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5


(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9


0

(1μV 0V 0V)


vx

v y

(d)


Complexity 9


References































10 Complexity






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018



















4 Complexity

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(b)


ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(a)

ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(b)


22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(b)


Complexity 5

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(b)


(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

k

y max

18

15

9

3

ndash3

ndash92 3 4 5 6 7 8

(a)

k

ndash05

0


ndash05

0

05(10minus6 3 0)

Lyap

unov

expo

nent

s

ndash05

0

05(10minus6 0 0)

2 3 4 5 6 7 8

(b)


y

x

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

6

6

4

4

2

2

0

0

ndash2

ndash2

ndash4

ndash4ndash6

ndash6

(a)

y

x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)9

6

3

0

ndash3

ndash6


(b)

Figure 8 Continued

6 Complexity







RCdvxdt

vy + vz minusR

Rk1sin vy( )

RCdvydt

minus vx + vz

RCdvzdt


Rk2sin vx( )

(12)





y

x


(10minus6 0 0)10

10

5

5

0

0

ndash5

ndash5ndash10

ndash10

(c)

y

x


(10minus6 0 0)

18

18

12

12

6

6

0

0

ndash6

ndash6

ndash12

ndash12ndash18

ndash18

(d)


Complexity 7




C1

C2

C3Rk2

10n

ndash+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

+

10n

10n

10k

sin(x)m

sin(x)m

10k

10k 10k10k

10k

10k

10k

10k

Vy

Vz

10k

10k

U2

U4

U6U5

VxVV

V

10k

Rk1U1

U3

deg

deg

deg

deg deg

deg

deg

deg

deg

deg deg

deg

deg

deg

deg

degdeg


ndash15ndash15

75

0

15

75

v y

vx

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 0V 0V)

(1μV 6V 0V)

(1μV minus6V 0V)

(1μV 16V 0V)


(a)

ndash15ndash15

75

0

15

75

v y


(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 1V 0V)

(1μV minus1V 0V)

(1μV 14V 0V)

(b)


8 Complexity

5 Conclusion



Data Availability




Acknowledgments


6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4


vx

v y

2 4 6

(a)

9

45

ndash45


0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5


(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9


0

(1μV 0V 0V)


vx

v y

(d)


Complexity 9


References































10 Complexity






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018








22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8

P0

P1

P2

P3

P4

P5

P6

P7

P8

CH

k1

k 2

(b)


ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(a)

ndash12ndash12

ndash8

ndash4

0

4

8

12


y (0)

(b)


22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

8ndash03

ndash02

ndash01

0

01

02

03

04

05

k1

k 2

(b)


Complexity 5

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(b)


(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

k

y max

18

15

9

3

ndash3

ndash92 3 4 5 6 7 8

(a)

k

ndash05

0


ndash05

0

05(10minus6 3 0)

Lyap

unov

expo

nent

s

ndash05

0

05(10minus6 0 0)

2 3 4 5 6 7 8

(b)


y

x

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

6

6

4

4

2

2

0

0

ndash2

ndash2

ndash4

ndash4ndash6

ndash6

(a)

y

x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)9

6

3

0

ndash3

ndash6


(b)

Figure 8 Continued

6 Complexity







RCdvxdt

vy + vz minusR

Rk1sin vy( )

RCdvydt

minus vx + vz

RCdvzdt


Rk2sin vx( )

(12)





y

x


(10minus6 0 0)10

10

5

5

0

0

ndash5

ndash5ndash10

ndash10

(c)

y

x


(10minus6 0 0)

18

18

12

12

6

6

0

0

ndash6

ndash6

ndash12

ndash12ndash18

ndash18

(d)


Complexity 7




C1

C2

C3Rk2

10n

ndash+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

+

10n

10n

10k

sin(x)m

sin(x)m

10k

10k 10k10k

10k

10k

10k

10k

Vy

Vz

10k

10k

U2

U4

U6U5

VxVV

V

10k

Rk1U1

U3

deg

deg

deg

deg deg

deg

deg

deg

deg

deg deg

deg

deg

deg

deg

degdeg


ndash15ndash15

75

0

15

75

v y

vx

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 0V 0V)

(1μV 6V 0V)

(1μV minus6V 0V)

(1μV 16V 0V)


(a)

ndash15ndash15

75

0

15

75

v y


(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 1V 0V)

(1μV minus1V 0V)

(1μV 14V 0V)

(b)


8 Complexity

5 Conclusion



Data Availability




Acknowledgments


6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4


vx

v y

2 4 6

(a)

9

45

ndash45


0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5


(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9


0

(1μV 0V 0V)


vx

v y

(d)


Complexity 9


References































10 Complexity






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018








22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(a)

22

3

3

4

4

5

5

6

6

7

7

8

80

01

02

03

04

05

k1

k 2

(b)


(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

k

y max

18

15

9

3

ndash3

ndash92 3 4 5 6 7 8

(a)

k

ndash05

0


ndash05

0

05(10minus6 3 0)

Lyap

unov

expo

nent

s

ndash05

0

05(10minus6 0 0)

2 3 4 5 6 7 8

(b)


y

x

(10minus6 minus3 0)

(10minus6 3 0)(10minus6 0 0)

6

6

4

4

2

2

0

0

ndash2

ndash2

ndash4

ndash4ndash6

ndash6

(a)

y

x

(10minus6 minus3 0)

(10minus6 3 0)

(10minus6 0 0)9

6

3

0

ndash3

ndash6


(b)

Figure 8 Continued

6 Complexity







RCdvxdt

vy + vz minusR

Rk1sin vy( )

RCdvydt

minus vx + vz

RCdvzdt


Rk2sin vx( )

(12)





y

x


(10minus6 0 0)10

10

5

5

0

0

ndash5

ndash5ndash10

ndash10

(c)

y

x


(10minus6 0 0)

18

18

12

12

6

6

0

0

ndash6

ndash6

ndash12

ndash12ndash18

ndash18

(d)


Complexity 7




C1

C2

C3Rk2

10n

ndash+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

+

10n

10n

10k

sin(x)m

sin(x)m

10k

10k 10k10k

10k

10k

10k

10k

Vy

Vz

10k

10k

U2

U4

U6U5

VxVV

V

10k

Rk1U1

U3

deg

deg

deg

deg deg

deg

deg

deg

deg

deg deg

deg

deg

deg

deg

degdeg


ndash15ndash15

75

0

15

75

v y

vx

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 0V 0V)

(1μV 6V 0V)

(1μV minus6V 0V)

(1μV 16V 0V)


(a)

ndash15ndash15

75

0

15

75

v y


(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 1V 0V)

(1μV minus1V 0V)

(1μV 14V 0V)

(b)


8 Complexity

5 Conclusion



Data Availability




Acknowledgments


6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4


vx

v y

2 4 6

(a)

9

45

ndash45


0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5


(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9


0

(1μV 0V 0V)


vx

v y

(d)


Complexity 9


References































10 Complexity






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018














RCdvxdt

vy + vz minusR

Rk1sin vy( )

RCdvydt

minus vx + vz

RCdvzdt


Rk2sin vx( )

(12)





y

x


(10minus6 0 0)10

10

5

5

0

0

ndash5

ndash5ndash10

ndash10

(c)

y

x


(10minus6 0 0)

18

18

12

12

6

6

0

0

ndash6

ndash6

ndash12

ndash12ndash18

ndash18

(d)


Complexity 7




C1

C2

C3Rk2

10n

ndash+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

+

10n

10n

10k

sin(x)m

sin(x)m

10k

10k 10k10k

10k

10k

10k

10k

Vy

Vz

10k

10k

U2

U4

U6U5

VxVV

V

10k

Rk1U1

U3

deg

deg

deg

deg deg

deg

deg

deg

deg

deg deg

deg

deg

deg

deg

degdeg


ndash15ndash15

75

0

15

75

v y

vx

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 0V 0V)

(1μV 6V 0V)

(1μV minus6V 0V)

(1μV 16V 0V)


(a)

ndash15ndash15

75

0

15

75

v y


(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 1V 0V)

(1μV minus1V 0V)

(1μV 14V 0V)

(b)


8 Complexity

5 Conclusion



Data Availability




Acknowledgments


6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4


vx

v y

2 4 6

(a)

9

45

ndash45


0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5


(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9


0

(1μV 0V 0V)


vx

v y

(d)


Complexity 9


References































10 Complexity






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018











C1

C2

C3Rk2

10n

ndash+

ndash

+

ndash

+

ndash

+

ndash

+

ndash

+

10n

10n

10k

sin(x)m

sin(x)m

10k

10k 10k10k

10k

10k

10k

10k

Vy

Vz

10k

10k

U2

U4

U6U5

VxVV

V

10k

Rk1U1

U3

deg

deg

deg

deg deg

deg

deg

deg

deg

deg deg

deg

deg

deg

deg

degdeg


ndash15ndash15

75

0

15

75

v y

vx

(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 0V 0V)

(1μV 6V 0V)

(1μV minus6V 0V)

(1μV 16V 0V)


(a)

ndash15ndash15

75

0

15

75

v y


(1μV minus3V 0V)

(1μV 3V 0V)

(1μV 1V 0V)

(1μV minus1V 0V)

(1μV 14V 0V)

(b)


8 Complexity

5 Conclusion



Data Availability




Acknowledgments


6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4


vx

v y

2 4 6

(a)

9

45

ndash45


0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5


(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9


0

(1μV 0V 0V)


vx

v y

(d)


Complexity 9


References































10 Complexity






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018








5 Conclusion



Data Availability




Acknowledgments


6

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

4

2

0

ndash2

ndash4


vx

v y

2 4 6

(a)

9

45

ndash45


0

(1μV 0V 0V)

(1μV 3V 0V)

(1μV ndash3V 0V)

vx

v y

(b)

10

5

0

ndash5


(1μV 3V 0V)

(1μV 0V 0V)

(1μV ndash3V 0V)

vx

v y

(c)

18

9

ndash9


0

(1μV 0V 0V)


vx

v y

(d)


Complexity 9


References































10 Complexity






















Complexity 11








Journal of












Journal of







Volume 2018






Volume 2018









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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