Symmetric coexisting attractors in a novel memristors-based Chua's chaotic system
EEPJ manuscript No. (will be inserted by the editor)
Symmetric coexisting attractors in a novel memristors-basedChuas chaotic system
Shaohui. Yan a , Zhenlong. Song, Wanlin. Shi, Weilong. Zhao, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou, Gansu 730070,ChinaReceived: date / Revised version: date
Abstract.
In this paper, based on the classic Chuas circuit, a charge-controlled memristor is introduced todesign a novel four-dimensional chaotic system. The complex dynamics of the novel chaotic system suchas equilibrium points, stability, dissipation, bifurcation diagrams, Lyapunov exponent spectra and phaseportraits are investigated. By varying the initial conditions of the system, it is found from numerical simu-lations that the system shows some dynamics of great interests including double-wings chaotic attractors,coexisting periodic-chaotic bubbles, asymmetric and symmetric coexisting attrators. The results show thatthe novel circuit system has extreme multistablity.
PACS.
Chuas Chaotic system Memristor Bifurcation analysis Coexisting attractors Symmetry
Nonlinear electronic circuits provide an effective wayto produce chaotic behavior [1] [2] [3]. Chuas circuit isa simple nonlinear chaotic circuit made by Professor CaiShaotang in1983 [4] [5] [6] [7]. Chua’s circuit contains fourbasic elements and a nonlinear resistance, but there havebeen hundreds of research papers. The details of Chua’scircuit have been deeply investigated including topology,numerical simulation, dynamical characterizations and phys-ical phenomena [8] [9] [10] [11] [12]. Because of the Chua’scircuit system has the characteristics of extreme initialvalue sensitivity and good pseudo-randomness, which hasbeen widely used in science and engineering, [13], robotics[14], random generator implementation [15], secure com-munication and even image encryption [16], and synchronousencryption [17]. The coexistence of multiple attractorshave been found in many of nonlinear systems and elec-tronic circuits [18] [19] [20] [21]. In general, the appear-ance of coexisting attractors is associated with systemssymmetry and depends closely on system initial condi-tions. Chaotic system with multiple attractors is able todeliver more complexity in chaos-based engineering ap-plications such as neural networks [22], image encryp-tion [23], control system [24] and random number gen-erator [25]. Therefore, chaotic system with coexisting at-tractors has become a considerable interest at present.In 1971, according to the completeness principle of cir-cuit theory Chua predicted the fourth electronic compo-nent and named memristor, which has the unique prop-erty of remembering the past electric charge [26] [27]. Thememristor was created by Hewlett Packard laboratory, a mortal [email protected] whose resistance was characterized by the nonlinear con-stitutive relation between charge and flux [28]. Becauseof nature non-linearity, plasticity of memristors, simplecircuit topology and complex dynamical behaviors, a lotof attraction have been attention to memristor-based ap-plications [29] [30], especially neural networks [31], write/ read circuits [32], image encryption [33], voice encryp-tion [34]. From the above discussion, these studies mainlyfocus on the using of flux-controlled memeristor. In 2008,HP laboratory announced the charge-controlled memris-tor, which is shown more practical application than theflux-controlled memristor. However,the charge-controlledmemristors have been little reported in the literature. Thus,chaotic system with the charge-controlled memristors hasimportant research value in the dynamics with the vari-ation of parameters and the multistability phenomenondepending on different initial conditions. This paper is or-ganized as follows: In Section 2. the mathematical modelof the Chua’s deformation circuit is completed. In Section3, the complex dynamical behaviors are numerically re-vealed by means of bifurcation diagrams, Lyapunov expo-nents, phase portraits and symmetric coexisting attractorsbehaviors. By changing the initial value, the Lyapunov ex-ponents actually show the symmetry of zero point. Finally,the conclusion is given in Section 4. In this section, a memristor-based chaotic circuit systemis introduced and its mathematical model is discussed. a r X i v : . [ n li n . C D ] S e p Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle
A floating ground type memristor chaotic circuit is shownin fig.1. Its specific structure design is used a nonlinearmemristor and an inductor in series between two capaci-tors, forming a connection method that allows the mem-ristor to float the ground type [35]. The memristive circuitalso contains one linear resistor R and one negative con-ductance minus G. The memristor model employs charge-controlled piece wise linear (PWL) model, the nonlinearrelationship between the variable q is shown (1). The cor-responding memristor value is defined as (2). Where q is the charge, M ( q ) are the charge-controlled resistancevalue, n is the resistance of memristor with q greater than1 and m is the resistance of memristor with q less than orequal to 1. f ( q ) = nq + 0 . m − n )( | q + 1 | − | q − | ) (1) M ( q ) = df ( q ) dq (cid:40) m | q | (cid:54) n | q | > A novel chaotic circuit based on the proposed memris-tor is constructed as showed fig.1. By applying Kirchhoffslaws, the circuit nonlinear differential equations can bederived as follows dv dt = 1 c ( i L + G N v ) dv dt = 1 c ( − v − i L ) di L dt = 1 L ( v − v − i M ( q )) dqdt = i l (3)By normalizing the state variables and circuit parametersare set as x = v , y = v , z = i L , u = q, α = 1 C , β = G N C , e = 1 L (4)among them M ( u ) = (cid:40) m | q | (cid:54) n | q | > ˙ x = αz + βx ˙ y = − γy − δz ˙ z = e ( y − x − M ( u ) z )˙ u = z (6)It implies that (6) is four-dimensional memristive chaoticsystem and has four state variables. The system param-eters are chosen as follows, α = 9 , β = 3 , γ = 4 . , δ = 15 , e = 0 . , m = 0 . , n = − . The equilibrium point of the mathematical model (6)can be expressed as p = ( x, y, z, u )whose values are solvedby the following equations: αz + βx − γy − δz e ( y − x − M ( u ) z )0 = z (7)Apparently, the equilibrium point can be gotten by setting A = { ( x, y, z, u ) | x = y = z = 0 , u = γ } , where (cid:48) γ (cid:48) isa constant. It means that this memristive chaotic systemhas a line of equilibrium corresponding to the u-axis. TheJacobian matrix in the equilibrium point can be easilygiven as: P ( λ ) = det t (1 λ − J ) = λ + Aλ + Bλ + C (8)Where A = γ + M ( u ) e − β, B = 2 αe + γM ( u ) e, C = e ( γM ( u ) β + αβ − γα )(9)According to the characteristic equation and known pa-rameters α = 9 , β = 3 , γ = 4 . , δ = 15 , e = 1 , m =0 . , n = − .
5, when | u | (cid:54) λ = 0 , λ = 0 . , λ , λ = − . ± . i (10)The root of λ is positive real constant, whereas the λ and λ are a pair of imaginary roots, which indicate that (10) isan unstable saddle focus with an index of 1. Because theyhave two complex conjugate roots with negative real partand one positive root [36]. When | u | > λ = 0 , λ = − . , λ , λ = 0 . ± . i (11)Also, the eigenvalues is an unstable saddle focus with anindex of 2. The saddle focus equilibrium of index 2 is thepremise for the generation of vortex motion, while thesaddle focus equilibrium of index 1 is the basis for theformation of bond bands between the coils [37]. Therefore,it can be concluded that the equilibrium point is alwaysunstable and conforms to the conditions for the generationof chaos [38]. lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle For the memristive chaotic system (6), its vector fielddivergence is: ∇ V = d ˙ xdx + d ˙ ydy + d ˙ zdz + d ˙ udu = β − γ − M ( u ) e (12)Fixing α = 9 , β = 3 , γ = 4 . , δ = 15 , m = 0 . , n = − . ande = 1 .
5, the ∇ V is obviously negative, when set-ting the | u | (cid:54) ∇ V = (cid:40) − . | u | (cid:54) . | u | > ∇ V = (cid:40) − . − . e | u | (cid:54) − . . e | u | > e is a variable parameter, it can obtain a diver-gence relation (14). Based on the analysis above, diver-gence relation of the system is specified for table.2. Thesystem is dissipative when | u | ≤ e > −
2, whichis also dissipative when | u | > e < /
15. Ideally,the phase space orbit after many folds and stretches caneventually be confined to a limited subset and eventuallyfixed in an attractive domain forming an attractor. Thenanother type of transition changes in the number of attrac-tors wings occur in this system. As e goes on, the phaseportrait transfers from double-wings becoming single-wingattractors, as shown in fig.3. In this subsection, the dynamic characteristics of sys-tem (6) are presented through bifurcation diagram, Lya-punov exponents spectrum and phase diagrams. The co-existing attractors mean multiple attractors with theirown domains of attraction with respect to different ini-tial variable. The numerical calculations are performedby Adomian decomposition method (ADM) with a fixedstep size is 0.01s [39]. For α = 9 , β = 3 , γ = 4 . , e =0 . , m = 0 . , n = − .
5, the bifurcation and Lyapunov ex-ponents diagram of system (6) with respect to parameter δ ∈ { , } starting from the initial value (x(0), y(0), z(0),u(0)) = (0.001, 0.02, 0, 0.1)) are shown in fig.4, whichshows the system is a weak hyperchaotic behaviors. Inthe hyperchaotic discontinuity region, the maximum Lya-punov index LE1 is always greater than zero, while LE2keeps jumping between greater than zero and less thanzero in fig.4(b). When > .
42, the dynamic behaviorof the system (6) shifts from chaos to periodic behaviorand then to the limit cycle with the evolution of δ . When δ = 17 .
33, it can be seen from the bifurcation diagram infig.4(a) that the system enters reverse periodic doublingbifurcation, which is well confirmed on the Lyapunov ex-ponents spectrum. When δ ∈ { , . } (except some nar-row periodic windows), the maximum Lyapunov exponent is not positive and the system is periodic orbit. When δ ∈ { . , . } , the maximum Lyapunov exponent ispositive, so the system is chaotic. When δ ∈ { . , } ,the maximum Lyapunov exponent is approximately zero,which means the system returns periodic orbit.As shown in fig.4(a), the points of A, B, C and D corre-spond to the phase portraits of fig.5.(a), fig.5(b), fig.5(c)and fig.5(d), respectively. A is Hop-bifurcation (HB), Bis hyperchaos interval(HI), C is reversed period doublingbifurcated point (RPDB), D is starting point of quasi-periodic (QP) and E is the point where the double-wingsattractor becomes single-wing attractor (DWA-SWA). Forthe fixed parameter of δ , the phase portraits demonstrat-ing different dynamical behaviors are provided, as shownin fig.5. fig.5(a) illustrates single-wing periodic state at δ =14 .
5. fig.5(b) illustrates double-wings asymmetric attrac-tors at δ = 14 .
8. fig.5(c) illustrates quasi-periodic stateat δ = 17 .
33, which shows that the window period exists.fig.5(d) illustrates limit cycle at δ = 17 .
65. The results infig.5. are consistent with those in fig.4 The system is ex-treme multistability, namely, the coexistence of attractorsfor different system parameter is distinctly observed.To investigate the effect of parameter on the dynam-ical behaviors of system (6), we let β = 3 , γ = 4 . , m =0 . , n = − . , δ = 15 , e = 0 .
91 and α ∈ { , } . Here, theinitial values are chosen as X0 (0.001, 0.02, 0, 0.1) and thecorresponding bifurcation diagram and Lyapunov expo-nents spectrum are shown in fig.6(a) and fig.6(b), respec-tively. Clearly, the maximum Lyapunov exponent is posi-tive and the system can generate chaos for α ∈ { , . } ∪{ . , . }∪{ . , } . For the phase diagram of fig.7(a),fig.7(b) and fig.7(c), there apparently emerge some pe-riodic windows sandwiched in the chaotic band. Whilethe system is periodic for α ∈ { . , . } ∪ { . , . } ∪{ , } ( as shown in fig.7(d), fig.7(e) and fig.7(f), respec-tively). There is a absorbing phenomenon in this chaotic sys-tem that has hardly been proposed in other memristivechaotic systems. It has shown that single-wing attractorand double-wings attractor appear to symmetrical rota-tion in this paper. The mentioned single-wing and double-wings attractor are aimed at the x − y phase diagrams.That is a single-wing attractor in one position can be sym-metrical rotated to generate single-wing attractor in dif-ferent locations. Usually, varying the parameters of systemmay induce the phenomenon of rotation. However, the ro-tation in this system is caused by different memristor ini-tial conditions, which is really different from the previousliteratures. Therefore, this phenomenon can be called therotation of coexisting attractors.Setting system parameters as α = 9 , β = 3 , γ = 4 . , e =0 . , m = 0 . , n = − . , δ = 15 and the initial conditionof three state variables as x (0) = 0 . , y (0) = 0 . z (0) = 0, then the memristor initial condition u (0) is takenas the Lyapunov exponent spectra parameter, as shownin fig.8(a). It can be acquired that the system undergoes Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle a complex alternation of numerous chaotic and periodicstates with u (0) increasing from -1 to 1. For the sake ofclarity, the middle complex part of the Lyapunov exponentspectra diagram is magnified, as exhibited in fig.8(b).Obviously, system (6) is symmetric in u − z axis. Con-sequently, a pair of symmetric attractors can be gener-ated by altering the initial conditions. Let α = 9 , β =3 , γ = 4 . , m = 0 . , n = − . , δ = 14 and initial values X . , . , , u ) and X . , . , , − u ). The sys-tem coexists two chaotic attractors, when parameters set e = 1 and u = 0 . e = 0 . u = 0 .
1, the system coexists one periodic-2 attractors withrespect to initial value X0 (blue) and X1 (red) as shownin fig.9(c) and fig.9(d). If e = 0.9 and u = 0.37, the systemcoexists one periodic-1 attractors and one chaotic attrac-tors with respect to initial value X0 (blue) and X1 (red) asshown in fig.9(e) and fig.9(f). If e = 1 , u = 0 .
37, the sys-tem is quasi periodic with respect to initial value X0 (blue)and X1 (red) as shown in fig.9(g) and fig.9(h). And theperiodic attractors place on symmetrical position of thephase space with similar structure. As described above,it shows that the novel chaotic system exists coexistingattractors and manifests the presence of multistability.
In this paper, we proposed the charge-controlled mem-ristor of chaotic circuit system. The dynamical behaviorsof the novel system are highly complex and sensitive withregard to different circuit parameters and initial condi-tions. Through using nonlinear analysis methods, it wasfound that some complex dynamical phenomena includ-ing state transition, chaos generation and degradation,and asymmetrical and symmetrical coexisting attractors.Simultaneously, we presented reasonable explanations forthese dynamical behaviors. Hence, the novel chaotic sys-tem has great prospects in engineering applications suchas image encryption, secure communication and neuralnetworks.
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Circuit schematic of the floating ground type memris-tive chaotic circuit
Fig. 2.
Phase portraits in (a) x − z plane, (b) x − y plane, (c) x − u plane, (d) the time-domain waveform of state variable x ,with the parameters α = 9 , β = 3 , γ = 4 . , δ = 15 , e = 0 . , m =0 . , n = − . x (0) = 0 . , y (0) =0 . , z (0) = 0 , u (0) = 0 . . Table 1.
The fixing control parameters for simulations.Parameters Values α β γ e m n -1.5 Table 2.
Divergence relation of system (6).Parameters u Parameters e divergence type | u | ≤ e > − (cid:53) V < | u | ≤ e < − (cid:53) V > | u | > e > (cid:53) V > | u | > e < (cid:53) V <
Fig. 3.
Two and three dimensional phase portraits with dif-ferent , (a) e = 1 .
5, Row 1 is the phase portrait in the x − y − z plane; Row 2, column 1 is the phase portrait in the x − u plane;Row 2, column 2 is the phase portrait in the x − y plane. (b) e = 1 .
5, Row 1 is the phase portrait in the x − y − z plane; Row2, column 1 is the phase portrait in the x − u plane; Row 2, col-umn 2 is the phase portrait in the x − y plane. The parametersare chosen as α = 9; β = 3; γ = 4 . , δ = 15 , m = 0 . , n = − . x (0) = 0 . , y (0) = 0 . , z (0) =0 , u (0) = 0 . . Fig. 4.
The diagram of (a) bifurcation and (b) Lyapunov ex-ponents with 14 δ
18. The parameters are chosen as α = 9 , β =3 , γ = 4 . , e = 0 . , m = 0 . , n = − . x (0) = 0 . , y (0) = 0 . , z (0) = 0 , u (0) = 0 . . Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle
Fig. 5.
Two and three dimensional phase portraits with dif-ferent δ . (a) δ = 14 .
5, Row 1, column 1 is the phase portraitin the x − y plane; Row 1 column 2 is the phase portrait inthe x − z plane; Row 2, column 1 is the phase portrait in the u − z plane; Row 2, column 2 is the phase portrait in the x − y − z plane. (b) δ = 14 .
8, Row 1, column 1 is the phaseportrait in the x − y plane; Row 1, column 2 is the phase por-trait in the x − z plane; Row 2, column 1 is the phase portraitin the u − z plane; Row 2, column 2 is the phase portrait inthe x − y − z plane. (c) δ = 17 .
33, Row 1, column 1 is thephase portrait in the x − y plane; Row 1, column 2 is thephase portrait in the x − z plane; Row 2, column 1 is the phaseportrait in the u − z plane; Row 2, column 2 is the phase por-trait in the x − y − z plane. (d) δ = 17 .
6, Row 1, column 1is the phase portrait in the x − y plane; Row 1, column 2 isthe phase portrait in the x − z plane; Row 2, column 1 is thephase portrait in the u − z plane, Row 2, column 2 is the phaseportrait in the x − y − z plane. The parameters are chosen as α = 9 , β = 3 , γ = 4 . , δ = 15 , m = 0 . , n = − . x (0) = 0 . , y (0) = 0 . , z (0) = 0 , u (0) = 0 . Fig. 6.
Corresponding bifurcation diagram (a) and Lyapunovexponents spectrum (b) with respect to parameter α . Fix β =3 , γ = 4 . , m = 0 . , n = − . , δ = 15 and e = 0 .
9, the initialvalues are x (0) = 0 . , y (0) = 0 . , z (0) = 0 , u (0) = 0 . . Fig. 7. phase portraits obtained for different values of c in: (a)hyperchaotic attractor at c = 9 .
19, (b) chaotic attractor at c =10 .
22, (c) chaotic attractor at c = 10 .
7, (d) quasi periodic at c = 9 . c = 10 .
41, (f) Limit cycle withperiod-1 at c = 11 .
7. The parameters are chosen as α = 9 , β =3 , γ = 4 . , δ = 15 , m = 0 . , n = − . x (0) = 0 . , y (0) = 0 . , z (0) = 0 , u (0) = 0 . . Fig. 8. symmetry Lyapunov exponent diagram. (a) Lyapunovexponent spectra, the parameter u (0)=-1 to 1. (b) Lyapunovexponent spectra, the parameter u (0)=-0.5 to 0.55. Fix β =3 , γ = 4 . , m = 0 . , n = 1 . , δ = 15 ande = 0 .
9, the initialvalues are x (0) = 0 . , y (0) = 0 . , z (0) = 0 , u (0).lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle Fig. 9.
Phase plane orbits in u-z plane and the correspond-ing time series of system (6) with parameters α = 9 , β =3 , γ = 4 . , m = 0 . , n = − . , δ = 14 , X . , . , , u ) and X . , . , , − u ). (a) coexisting chaotic attrators and (b)time series of u at e = 1 , u = 0 .
1, (c) coexisting periodic-2 at-tractors (d) time series of u at e = 0.9, u = 0.1, (e) coexistingperiodic-1 and (f) time series of u at e = 0 . .u = 0 .
37, (g)quasi periodic at and (h) time series of u at e = 1 , u = 0 ..