五阶压控忆阻蔡氏混沌电路的双稳定性
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摘要
通过在蔡氏电路的耦合电阻支路中串联一个电感,采用压控忆阻替换蔡氏电路中的蔡氏二极管,提出了一种新颖的五阶压控忆阻蔡氏混沌电路.建立该电路的数学模型,从理论上分析了平衡点及其稳定性的演化过程.特别地,该电路在给定参数下只有一个不稳定的零平衡点,却形成了混沌与周期的非对称吸引子共存的吸引盆,意味着双稳定性的存在.进而利用数值仿真与PSIM电路仿真着重研究了本文电路在不同初始状态下产生的双稳定性现象及其形成机理. PSIM电路仿真结果与数值仿真结果一致,较好地验证了理论分析.借助分岔图、李雅普诺夫指数、相轨图和吸引盆进一步深入探讨了归一化五阶压控忆阻蔡氏系统依赖于系统初始条件的动力学行为.结果表明,该忆阻蔡氏系统在不同的初始条件下能够呈现出混沌吸引子与周期极限环共存的双稳定性现象.
Generally, the occurrence of multiple attractors indicates that the multi-stability existing in a nonlinear dynamical system and the long-time motion behavior are essentially different, depending on which basin of attraction the initial condition belongs to. Up to now, due to the emergence of multi-stability, some particular memristor-based nonlinear circuits whose dynamical behaviors are extremely related to memristor initial conditions or other initial conditions have attracted considerable attention. By replacing linear or nonlinear resistors with memristor emulators in some already-existing oscillating circuits or introducing memristor emulators with different nonlinearities into these oscillating circuits, various memristor-based nonlinear dynamical circuits have been constructed and broadly investigated. Motivated by these considerations, we present a novel fifth-order voltage-controlled memristor-based Chua's chaotic circuit in this paper, from which a wonderful phenomenon of bi-stability is well demonstrated by numerical simulations and PSIM circuit simulations. Note that the bi-stability is a special kind of multi-stability, which is rarely reported in the memristor-based chaotic circuits.The proposed memristor-based Chua's chaotic circuit is constructed by inserting an inductor into the coupled resistor branch in series and substituting the Chua's diode with a voltage-controlled memristor in the classical Chua's circuit.Five-dimensional system model is established, of which the equilibrium point and its stability are investigated. Theoretical derivation results indicate that the proposed circuit owns one or three equilibrium points related to the circuit parameters.Especially, unlike the newly reported memristive circuit with bi-stability, the proposed memristor-based Chua's chaotic circuit has only one zero equilibrium point under the given parameters, but it can generate coexistent chaotic and periodic behaviors, and the bi-stability occurs in such a memristive Chua's circuit. By theoretical analyses, numerical simulations and PSIM circuit simulations, the bi-stability phenomenon of coexistent chaotic attractors and periodic limit cycles with different initial conditions and their formation mechanism are revealed and expounded. Besides, with the dimensionless system equations, the corresponding initial condition-dependent dynamical behaviors are further numerically explored through bifurcation diagram, Lyapunov exponents, phased portraits and attraction basin. Numerical simulation results demonstrate that the proposed memristive Chua's system can generate bi-stability under different initial conditions.The PSIM circuit simulations and numerical simulations are consistent well with each other, which perfectly verifies the theoretical analyses.
Generally, the occurrence of multiple attractors indicates that the multi-stability existing in a nonlinear dynamical system and the long-time motion behavior are essentially different, depending on which basin of attraction the initial condition belongs to. Up to now, due to the emergence of multi-stability, some particular memristor-based nonlinear circuits whose dynamical behaviors are extremely related to memristor initial conditions or other initial conditions have attracted considerable attention. By replacing linear or nonlinear resistors with memristor emulators in some already-existing oscillating circuits or introducing memristor emulators with different nonlinearities into these oscillating circuits, various memristor-based nonlinear dynamical circuits have been constructed and broadly investigated. Motivated by these considerations, we present a novel fifth-order voltage-controlled memristor-based Chua's chaotic circuit in this paper, from which a wonderful phenomenon of bi-stability is well demonstrated by numerical simulations and PSIM circuit simulations. Note that the bi-stability is a special kind of multi-stability, which is rarely reported in the memristor-based chaotic circuits.The proposed memristor-based Chua's chaotic circuit is constructed by inserting an inductor into the coupled resistor branch in series and substituting the Chua's diode with a voltage-controlled memristor in the classical Chua's circuit.Five-dimensional system model is established, of which the equilibrium point and its stability are investigated. Theoretical derivation results indicate that the proposed circuit owns one or three equilibrium points related to the circuit parameters.Especially, unlike the newly reported memristive circuit with bi-stability, the proposed memristor-based Chua's chaotic circuit has only one zero equilibrium point under the given parameters, but it can generate coexistent chaotic and periodic behaviors, and the bi-stability occurs in such a memristive Chua's circuit. By theoretical analyses, numerical simulations and PSIM circuit simulations, the bi-stability phenomenon of coexistent chaotic attractors and periodic limit cycles with different initial conditions and their formation mechanism are revealed and expounded. Besides, with the dimensionless system equations, the corresponding initial condition-dependent dynamical behaviors are further numerically explored through bifurcation diagram, Lyapunov exponents, phased portraits and attraction basin. Numerical simulation results demonstrate that the proposed memristive Chua's system can generate bi-stability under different initial conditions.The PSIM circuit simulations and numerical simulations are consistent well with each other, which perfectly verifies the theoretical analyses.
引文
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[31] Kengne J, Tabekoueng Z N, Tamba V K, Negou A N2015 Chaos 25 103126
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[33] Feudel U 2008 Int. J. Bifurcation Chaos 18 1607
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[35] Morfu S, Nofiele B, MarquiéP 2007 Phys. Lett. A 367192
[36] Bao B C, Xu J P, Zhou G H, Ma Z H, Zou L 2011 Chin.Phys. B 20 120502
[37] Chua L O 2012 Proc. IEEE 100 1920
[38] Ma J, Chen Z Q, Wang Z L, Zhang Q 2015 Nonlinear Dyn. 81 1275
[39] Bao B C, Jiang T, Xu Q, Chen M, Wu H G, Hu Y H2016 Nonlinear Dyn. 86 1711
[40] Wolf A, Swift J B, Swinney H L, Vastano J A 1985Physica D 16 285
[2] Strukov D B, Snider G S, Stewart D R, Williams R S2008 Nature 453 80
[3] Kim H, Sah M P, Yang C, Cho S, Chua L O 2012 IEEE Trans. Circuits Syst. I:Regular Papers 59 2422
[4] Yu D S, Iu H H C, Fitch A L, Liang Y 2014 IEEE Trans.Circuits Syst. I:Regular Papers 61 2888
[5] Sánchez-López C, Mendoza-López J, Carrasco-Aguilar M A, Mu?iz-Montero C 2014 IEEE Trans. Circuits Syst.II:Express Briefs 61 309
[6] Yang C, Choi H, Park S, Sah M P, Kim H, Chua L O2015 Semicond. Sci. Technol. 30 015007
[7] Yu D S, Zheng C Y, Iu H H C, Fernando T, Chua L O2017 IEEE Access 5 1284
[8] Corinto F, Ascoli A 2012 Electron. Lett. 48 824
[9] Bao B C, Yu J J, Hu F W, Liu Z 2014 Int. J. Bifurcation Chaos 24 1450143
[10] Wu H G, Bao B C, Liu Z, Xu Q, Jiang P 2016 Nonlinear Dyn. 83 893
[11] Xu Q, Lin Y, Bao B C, Chen M 2016 Chaos, Solitons Fractals 83 186
[12] Bao B C, Wang N, Xu Q, Wu H G, Hu Y H 2017 IEEE Trans. Circuits Syst. II:Express Briefs 64 977
[13] Xu Q, Zhang Q L, Bao B C, Hu Y H 2017 IEEE Access5 21039
[14] Bao H, Bao B C, Lin Y, Wang J, Wu H G 2016 Acta Phys. Sin. 65 180501(in Chinese)[包涵,包伯成,林毅,王将,武花干2016物理学报65 180501]
[15] Chen M, Yu J J, Bao B C 2015 Electron. Lett. 51 462
[16] Chen M, Li M Y, Yu Q, Bao B C, Xu Q, Wang J 2015Nonlinear Dyn. 81 215
[17] Bao B C, Hu F W, Liu Z, Xu J P 2014 Chin. Phys. B23 070503
[18] Bao B C, Jiang P, Wu H G, Hu F W 2015 Nonlinear Dyn. 79 2333
[19] Fitch A L, Yu D S, Iu H H C, Sreeram V 2012 Int. J.Bifurcation Chaos 22 1250133
[20] Yang F Y, Leng J L, Li Q D 2014 Acta Phys. Sin. 63080502(in Chinese)[杨芳艳,冷家丽,李清都2014物理学报63 080502]
[21] Wu H G, Chen S Y, Bao B C 2015 Acta Phys. Sin. 64030501(in Chinese)[武花干,陈胜垚,包伯成2015物理学报64 030501]
[22] Bao B C, Ma Z H, Xu J P, Liu Z, Xu Q 2011 Int. J.Bifurcation Chaos 21 2629
[23] Li Q D, Zeng H Z, Li J 2015 Nonlinear Dyn. 79 2295
[24] Zhao Y B, Zhang X Z, Xu J, Guo Y C 2015 Chaos,Solitons Fractals 81 315
[25] Bao B C, Hu F W, Chen M, Xu Q, Yu Y J 2015 Int. J.Bifurcation Chaos 25 1550075
[26] Li C B, Sprott J C 2014 Int. J. Bifurcation Chaos 241450034
[27] Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77255
[28] Wei Z C, Wang R R, Liu A P 2014 Math. Comput. Simulat. 100 13
[29] Bao B C, Wu H G, Xu L, Chen M, Hu W 2018 Int. J.Bifurcation Chaos 28 1850019
[30] Richter H 2008 Chaos, Solitons Fractals 36 559
[31] Kengne J, Tabekoueng Z N, Tamba V K, Negou A N2015 Chaos 25 103126
[32] Bao H, Wang N, Wu H G, Song Z, Bao B C 2018 IETE Tech. Rev. 6 1
[33] Feudel U 2008 Int. J. Bifurcation Chaos 18 1607
[34] Chen M, Sun M X, Bao B C, Wu H G, Xu Q, Wang J2018 Nonlinear Dyn. 91 1395
[35] Morfu S, Nofiele B, MarquiéP 2007 Phys. Lett. A 367192
[36] Bao B C, Xu J P, Zhou G H, Ma Z H, Zou L 2011 Chin.Phys. B 20 120502
[37] Chua L O 2012 Proc. IEEE 100 1920
[38] Ma J, Chen Z Q, Wang Z L, Zhang Q 2015 Nonlinear Dyn. 81 1275
[39] Bao B C, Jiang T, Xu Q, Chen M, Wu H G, Hu Y H2016 Nonlinear Dyn. 86 1711
[40] Wolf A, Swift J B, Swinney H L, Vastano J A 1985Physica D 16 285