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Dynamical Behaviors of A Modified Chuas Circuit 1 School of Software and Information Engineering, University of Electronic Science and Technology of China, Chengdu, 610054, China2 School of Information Science and Technology, Tibet University, Lhasa, 850000, China Higher Education Press and Springer-Verlag Berlin Heidelberg 2017AbstractA modified Chuas circuit based on meminductor and the improve memristive diode bridge emulator is implemented in this paper. The state equations of the modified Chuas circuit are described, and the dynamical behaviors such as equilibrium set, Lyapunov exponents and diagram are revealed by theoretical analyses 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 the coexisting attractors depending on the system initial value and parameters. Furthermore, the simulation results demonstrate that some strange chaotic attractors such as double-scroll type and like butterfly attractor are displayed when the parameter and initial value changed. 1 IntroductionIn 1971, Leon Chua proposed the fourth fundamental passive element which called memristor 1, it has not been fabricated until 2008 2. In 2009, Ventra M.D extends 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 advance 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 propose several nonlinear oscillators in 2008 11. Muthuswamy present a simplest chaotic circuit that including three elements of the memristor, inductor and capacitor in the reference 12. Fang Yuan and Gangyi Wang proposed a meminductor-based circuit in the 2016 13. Bao Bocheng etc. presented a simple third-order memristive band pass filter chaotic 14, non-autonomous seconed-order memristive chaotic circuit 15, and memristor-based Wien-bridge oscillator 16,17and so on. Reference 18-19 presented chaotic circuit based on meminductor and memcapacitor, respectively. However, as we known, there are few papers have reported chaotic circuit based on memristor and memcapacitor. Furthermore, many researchers have analyzed complex dynamic behaviors of chaotic circuits. For example, coexisting attractors are analyzed in the references 14,20,21, state transition is reported in 22, multi hidden attractors are studied in 23. In this paper, we focus on the construction 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, bifurcation diagram. It is worthy of notice that the coexisting attractors depending the system initial value and parameters are studied.The rest of this paper is organized in the following manner. The section 2 mainly proposed a novel Chuas circuit based on meminductor and the improve memristive diode bridge emulator and introduced the prerequisite knowledge. Section 3 mainly analyze the equilibrium set of the chaotic circuit, experimental results and dynamical behavior of the circuit are provided in section 4. Finally, the conclusions are drawn in section 6.2 Modeling of the Improved Chuas CircuitFigure.1 the improve memristive diode bridge emulatorFigure. 2 The modified Chuas circuitThe modified Chuas circuit is presented in this paper, as shown in Figure 1, By replacing the Chuas diode and inductor with improve memristive diode bridge emulator and meminductor. Herein, the improve memristive diode bridge emulator is reported in the 24, the circuit scheme of the improve memristive diode bridge emulator shown in Figure. 1, v and i are the input voltage and current of the improve memristor diode bridge emulator, respectively. iL is the current of the inductor, the mathematical model as described, (1) (2)where,IS, n and VT are the internal state of diode.A 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 constant, .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) (6)Denote x=v1, z=v2, y=L1, 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)When the parameters are given by a=1, b=0.006, c=1, d=1, G=1, m=-2.83, n=6.9, 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 Figure.23 Equilibrium and its stabilityThrough setting the of equation (6), the equilibrium set of the system can be described asWhere k is real constant. The Jacobian matrix at this equilibrium set of memristive elements circuit can be derived as4 Dynamical behavior of the a modified Chuas circuit 4.1. Parameter Dependent Bifurcation BehaviorsWith the variation of the parameter f, the bifurcation diagram and first three maximum Lyapunov exponents are plotted in Figs.3, it can be observed that the chaotic circuit have 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 f1goes along in the region 0.008,0.034. In the region of the f goes along (0.034,0.048, some periodic windows with different periodicities occur. Finally, the system enters into period at f=0.048. Figs. 4 shows a variety of phase diagrams when f takes different values. Table I described (a) and Figs.4(b) shows the double scroll chaotic attractor, Figs.4(c) shows the 3D chaotic attractor. In is noted that the circuit performs. Figs.3 For the initial values (0.224 0.2024 0.1 -0.000154 -1.2294). (a) Bifurcation diagram of f1. (b)First three Lyapunov exponents.4.2 Multipe coexisting attractors depending on x(0)Consider that the x(0) increases in the region 2,2, the system(6) parameters 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 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) in the Figs.4(a) and the corresponding three Lyapunov exponents are shown in Figs.4(b) and Figs.4(c). In the region 2, 0.25 and 0.25,2, the system(6) displayed the coexisting period and strange attractors, as shown in Figs.5(a) and Figs.5(b). However, in the region(0.25,0.25), the system occurred chaotic behaviors, as shown in Figs.5(c). It is noted that the crisis scenarios happen at x(0)= 0.25.Fig.4.Bifurcation diagrams of x and first three exponents with the initial x(0) increasing. (a) the bifurcation diagrams. (b) and (c) the Lyapunov exponents6. ConclusionsIn this paper, a modified Chuas circuit based on meminductor and the improve memristive diode bridge emulator is implemented in this paper. the nonlinear dynamic behaviors of the chaos circuit by using the chaos attractor, equilibrium set, Lyapunov exponent, bifurcation diagram are been analyzed. Simulation result shows that this circuit can produce chaotic oscillation, which can be applied into the encryption system and neural network. This type of curcuit will contribute to researching the physical characteristics of memory elements.References1. L. O. Chua, “MemristorThe missing circuit element,” IEEE Trans. Circuit Theory, vol. 18, no. 5, pp. 507519, Sep. 1971.2. D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, no. 7191, pp. 8083, 2008.3. M. Di Ventra, Y.V. Pershin, and L.O. Chua, “Circuit elements with memory: memristors, memcapacitors and meminductors”, Proc. IEEE, no.97, vol.10, pp.1717-1724, 2009.4. Y. Ho, G. M. Huang, and P. Li. “Nonvolatile memristor memory” device characteristics and design implications. In ICCAD 2009, Nov. 2009.5. Kavehei O, Iqbal A, Kim Y S, Eshraghian K, Al-Sarawi SFand Abbott D 2010 The fourth element: characteristics, modelling and electromagnetic theory of the memristor Proc. R. Soc. A 466 21752026. P. Mazumder, S. M. Kang, and R. Waser, Memristors: Devices, models, and applications, Proc. IEEE, vol. 100, no. 6, pp. 19111919, Jun. 2012.7. Hu M, Graves C E, Li C, et al. MemristorBased Analog Computation and Neural Network Classification with a Dot Product EngineJ. Advanced Materials, vol.30, no.9, 2018.8. Li B, Wang Y, Wang Y, et al. Training itself: Mixed-signal training acceleration for memristor-based neural networkC/ Asia and South Pacific Design Automation Conference. IEEE, pp.361-366, 2014.9. Sun, J.W., Shen, Y., Yin, Q., Xu, C.J.: Compound synchronization of four memristor chaotic oscillator systems and secure communication. Chaos 23, 013140 (2013)10. Li Y, Yang C, Yu Y. Research on low pass filter based on Memristor and Memcapacitor. CCC, 2017.11. Makoto Itoh, L. O. Chua, “Memristor oscillators,” International Journal of Bifurcation and Chaos, vol.18, pp. 3183-3206, 2008.12. B. Muthuswamy, “Implementing memristor based chaotic circuits,” Int. J. Bifurcation Chaos, vol. 20, no. 5, pp. 13351350, May 2010.13. Yuan F, Wang G, Jin P, et al. Chaos in a Meminductor-Based CircuitJ. International Journal of Bifurcation & Chaos, 2016, 26(08):1650130.14. Bao B, Wang N, Xu Q, et al. A Simple Third-Order Memristive Band Pass Filter Chaotic CircuitJ. IEEE Transactions on Circuits & Systems II Express Briefs, 2016, PP (99):1-1.15. Xu Q, Zhang Q, Bao B, et al. Non-Autonomous Second-Order Memristive Chaotic CircuitJ. IEEE Access, 2017, PP (99):1-1.16. Bao H, Wang N, Wu H, et al. Bi-Stability in an Improved Memristor-Based Third-Order Wien-Bridge OscillatorJ. Iete Technical Review, 2018(6):1-8.17. Wang N, Bao B, Jiang T, et al. Parameter-Independent Dynamical Behaviors in Memristor-Based Wien-Bridge Oscil