杭州论文 Dynamical Behaviors of A Modified Chua’s Circuit

Dynamical Behaviors of A Modified Chuas CircuitHuihui Ma, Yongbin Yu, Chenyu Yang, Nijing Yang, Yancheng Wang, and Xiangxiang WangSchool of Information and Software Engineering University of Electronic Science and Technology of ChinaChengdu, ChinaTashi NyimaSchool of Information Science and TechnologyTibet UniversityTibet, CAbstractA modified Chuas circuit is implemented meminductor and the improved memristive diode bridge emulator in this paper. The state equations of the modified Chuas circuit are described. The dynamical behaviors such as equilibrium set, Lyapunov exponents and diagram are revealed by theoretical analysis and numerical simulations. Furthermore, the circuit exhibits rich dynamical behaviors including chaos, hyperchaos, periodic windows, period states, crisis scenarios and coexisting attractors. It is noted that coexisting attractors depend on the initial values and parameters of system. Furthermore, the simulation results demonstrate that some strange chaotic attractors such as double-scroll type and like butterfly attractor are displayed when parameter and initial value are changing.Keywords; Chuas circuit; Dynamic;attractors ; simulationI. Introduction (Heading 1)In 1971, Leon Chua proposed the fourth fundamental passive element which was called memristor 1, it has not been fabricated until 2008 2. In 2009, Ventra M.D extended the concept of memristor to memcapacitor and meminductor 3. It is noted that memristor is a two terminal circuit element that has many unique properties, such as nonvolatility 4, nonlinearity 5, and nanometer geometries 2. Hence, memristor is widely applied to various fields like data storage 6, neural networks 7,8, secure communications 9, filter circuit 10 and so on. In particular, memristor-composed circuits are more likely to generate high frequency chaotic oscillation signals. Because of the controllable and nanoscale attributes of memristor, it can be used to stimulate the development of the nonlinear field.In recent years, many nonlinear researchers are devoted to studying chaotic circuit based on memristor, memcapacitor or meminductor. Itoh and Chua proposed several nonlinear oscillators in 2008 11. Muthuswamy presented a simplest chaotic circuit that includes three elements of the memristor, inductor and capacitor in the literature 12. Fang Yuan and Gangyi Wang proposed a meminductor-based circuit in 2016 13. Bao Bocheng et al. presented a simple third-order memristive band pass filter chaotic 14, non-autonomous seconed-order memristive chaotic circuit 15, memristor-based Wien-bridge oscillator 16,17and so on. Literature 18 and 19 presented chaotic circuit based on meminductor and memcapacitor respectively. However, there are few papers that have reported chaotic circuit based on memristor and memcapacitor. Furthermore, many researchers have analyzed complex dynamic behaviors of chaotic circuits. For example, coexisting attractors were analyzed in the literature 14,20,21, state transition was reported in 22, multi hidden attractors were studied in 23. In this paper, we focus on the construction of a novel Chuas circuit based on meminductor and the improve memristive diode bridge emulator. The complex nonlinear dynamic behaviors such as chaos, hyperchaos, periodic windows, period states, crisis scenarios and coexisting attractors are investigated by using the Phase Diagrams, equilibrium set, Lyapunov exponent, and bifurcation diagram. It is worthy of noticing that the coexisting attractors depending the system initial value and parameters are studied.The rest of this paper is organized as follows. Section proposes a novel Chuas circuit based on meminductor and the improve memristive diode bridge emulator and introduces the prerequisite knowledge. Section analyzes the equilibrium set of the chaotic circuit. Experimental results and dynamical behavior of circuit are provided in section . Finally, the conclusions are drawn in section .II. MODELING OF THE IMPROVED CHUAS CIRCUITThe modified Chuas circuit is presented in this paper, as shown in Figure 1, By replacing the Chuas diode and inductor with improved memristive diode bridge emulator and meminductor. Herein, the improved memristive diode bridge emulator is reported in literature 24. The circuit scheme of the improved memristive diode bridge emulator is shown in Figure. 1. v and i are the input voltage and current of the improved memristor diode bridge emulator respectively. iL is the current of inductor. The mathematical model is described asFigure.1 the improve memristive diode bridge emulatorFigure. 2 The modified Chuas circuit (1) (2)where,IS, n and VT are the internal states of diode.The smooth continuous cubic monotone-increasing nonlinearity meminductor 13 is described by (3)The derivative equation form (4) with respect to time t can be obtained (4)where c and d are constants, .The two models are applied in modified Chuas circuit.Figure.2 shows that the modified Chuas circuit contains two capacitor C1 and C2, a negative conductor G, a memristor MR and a meminductor ML. By applying Kirchhoffs laws to the circuit in Fig.2, the five state equations of the modified Chuas circuit are described by (5) Denote x=v1, y=L1, z=v2, w=iL, u=, ,a=1/C1, b=IS, c=, e=1/C2, f=L2The fifth order dimension-less mathematical model can be obtained by system(5) (6)When the parameters are given by a=1, b=0.005, c=1, d=1, G=1, m=3, n=7, f=0.005 and initial conditions (0.224, 0.2024,0.1, 0.000154, 1.2294), the double-scroll chaotic attractors can be exhibited as shown in Figs.3III. EQUILIBRIUM AND ITS STABILITYThrough setting the of equation (6), the equilibrium set of the system can be described as (7)where k is a real constant. The Jacobian matrix at this equilibrium set of a modified Chuas circuit can be derived as (8)Then the characteristic equation can be described as follows (9) whereAccording to the Routh-Hurwitz conditions, the set of inequalities are given by (10) If the set of inequalities (10) above are satisfied, the equilibrium point is stable, leading to the occurrence of point attractor. Whereas if any one of the four conditions is not satisfied, the equilibrium point is unstable which is the necessary condition for chaos. When the other parameters are unchanged, the corresponding stable regions in the parameter |k| is changed from 3.6048 to 26.8672.IV. DYNAMICAL BEHAVIOR OF A MODIFIED CHUAS CIRCUITA. Parameter Dependent Bifurcation BehaviorsWith the variation of the parameter f, the bifurcation diagram and first three maximum Lyapunov exponents are plotted in Figs.4. It can be observed that the chaotic circuit has rich dynamical features including chaos, hyperchaos, periodic windows and period states. When the f goes along in the regions 0.0001,0.008), the system shows chaotic behavior. the hyperchaos dynamic phenomena has occurred of f1 going along in the region 0.008,0.034. In the region of the f going along (0.034,0.048, some periodic windows with different periodicities occur. Finally, the system enters into period at f=0.048. Figs. 5 shows a variety of phase diagrams when f takes different values. Figs.5(a) and Figs.5(b) show the phase diagrams when system is in chaos. Figs.4(c) and Figs.4(d) show the phase diagrams when system is in hyperchaos. Figs.5(e) and Figs.5(f) show that the phase diagrams when system is in period. (a) (b)Figs.3 chaotic attractors of a modified Chuas circuit. (a)Y-U plane. (b) X-Y plane(a)(b)Figs.4 for the initial values (0.224 0.2024 0.1 -0.000154 -1.2294). (a) Bifurcation diagram of f1. (b)First three Lyapunov exponents. (a) (b) (c) (d) (e) (f)Figs.5 The different phase diagrams when f takes different values. (a) X-W plane when f=0.0005. (b) X-U-Y plane when the f=0.0005. (c)X-W plane when f=0.0134. (d)X-U-Y plane when f=0.0134. (e) X-W plane when f=0.06 (f)X-U-Y plane when f=0.06.B. Multiple coexisting attractors depending on x(0)Considering that the x(0) increases in the region 2,2, the parameters of system(6) are unchanged, and initial conditions are set as (x(0), 0.2024,0.1, 0.000154, 1.2294) and (x(0), 0.2024,0.1, 0.000154, 1.2294) respectively. The bifurcation diagram and first three maximum Lyapunov exponents are plotted in Figs.4. The red trajectories in the Figs.6(a) is when the initial value is(x(0), 0.2024,0.1, 0.000154, 1.2294) and the blue trajectories is when the initial value is(x(0), 0.2024,0.1, 0.000154, 1.2294). The corresponding three Lyapunov exponents are shown in Figs.6(b) and Figs.6(c). In the region 2, 0.25 and 0.25,2, the system(6) displays coexisting period and strange attractors, as shown in Figs.7(a) and Figs.7(b). However, in the region(0.25,0.25), the system occurred chaotic behaviors, as shown in Figs.7(c). It is noted that the crisis scenarios happen at x(0)= 0.25. (a)(b)(c)Fig.6.Bifurcation diagrams of x and first three exponents with the initial x(0) increasing.(a) the bifurcation diagrams. (b) and (c) the Lyapunov exponents. (a) (b) (c)Figs.7 the coexisting phase trajectories X-Y plane. (a). x(0)=1, (b). x(0)= 1, (c). x(0)=0.0224V. CONCLUSIONIn this paper, a modified Chuas circuit based on meminductor and an improved memristive diode bridge emulator is implemented. The nonlinear dynamic behaviors of the chaos circuit by using the chaos attractor, equilibrium set, Lyapunov exponent and bifurcation diagram have been analyzed. Simulation result shows that this circuit have generated rich dynamical behaviors including chaos, hyperchaos, periodic windows, period states, crisis scenarios and coexisting attractors with changes in parameters and initial values. Not only can this circuit be applied into the encryption system and neural network, but also be used to stimulate the development of the nonlinear field.AcknowledgmentThis work is supported by National Natural Science Foundation of China (NSFC Grant No.61550110248). The authors would like to thank the editor and the reviewers for their helpful suggestions and valuable comments.References1 L. O. Chua, “MemristorThe missing circuit element,” IEEE Trans. Circuit Theory, vol. 18, Sep. 1971, pp. 507519.2 D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, 2008, pp. 8083.3 M. Di Ventra, Y.V. Pershin, and L.O. Chua, “Circuit elements with memory: memristors, memcapacitors and meminductors”, Proc. 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