一类改进Chua系统隐藏吸引子研究
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摘要
对一类改进的Chua系统进行定位系统的隐藏吸引子研究。首先,通过对改进的Chua系统平衡点稳定性进行分析,确定系统在一定条件下有一对纯虚特征根。由于存在一对纯虚特征根,系统在平衡点处就会出现Hopf分支。然后,通过描述对原系统进行变换,以引入一系列连续函数序列对系统进行解析数值算法迭代,与定位稳定周期解的谐波线性化方法结合定位原系统的隐藏吸引子。通过MATLAB数学软件进行数值模拟得到系统的Lyapunov指数谱和分岔图,从而为隐藏吸引子的存在性提供依据。运用相关程序在MATLAB的帮助下制作出系统的隐藏吸引子相图,得出在这类改进的Chua系统中存在隐藏吸引子的结论。
Firstly,by discussing the stability of the equilibrium point of the improved Chua system,we can determine that the system has a pair of pure imaginary eigenvalues under certain conditions.By describing the transformation of the original system and introducing a series of continuous function sequences to iterate the analytical numerical algorithm of the system,and combining with the harmonic linearization method to locate the stable periodic solution,the hidden attractor of the original system is located.Then,the Lyapunov exponential spectrum and bifurcation diagram of the system is obtained by numerical simulation with MATLAB software,which provides the possibility for the existence of hidden attractor.Then,with the help of Matlab,the phase diagram of hidden attractor is obtained.Finally,it is concluded that hidden attractor exists in this improved Chua system.
Firstly,by discussing the stability of the equilibrium point of the improved Chua system,we can determine that the system has a pair of pure imaginary eigenvalues under certain conditions.By describing the transformation of the original system and introducing a series of continuous function sequences to iterate the analytical numerical algorithm of the system,and combining with the harmonic linearization method to locate the stable periodic solution,the hidden attractor of the original system is located.Then,the Lyapunov exponential spectrum and bifurcation diagram of the system is obtained by numerical simulation with MATLAB software,which provides the possibility for the existence of hidden attractor.Then,with the help of Matlab,the phase diagram of hidden attractor is obtained.Finally,it is concluded that hidden attractor exists in this improved Chua system.
引文
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[18] DUDKOWAKI D,JAFARI S,KAPITTANIAK T,et al.Hidden attractors in dynamical systems[J].Physics Reports,2016,637:1-50.
[19] LEONOV G A,KUZNETSOV N V.Hidden attractors in dynamical systems from hidden oscillations in Hilbert-Holmogorov,Aizerman,and Kalman problems to hidden chaotic attractor in Chua circuits[J].International Journal Of Bifurcation&Chaos,2013,23:1330002.
[20] TESI A,VICINO A.Robust absolute stability of lur'e control systems in parameter space[M].Pergamon Press,Inc.1991,27(1):147-151.
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[2] ROSSLER O E.An equation for continuous chaos[J].Physics Letters A,1976,57(5):397-398.
[3] CHUA L O,LIN G N.Canonical realization of chua's circuit family[J].IEEE Transactions on Circuits&Systems,1990,37(7):885-902.
[4] CHEN G R,UETA T.Yet another chaotic attractor[J].International Journal Of Bifurcation&Chaos,1999,9(7):1465-1466.
[5] QI G,CHEN G,DU S,et al.Analysis of a new chaotic system[J].Physica A Statistical Mechanics&Its Applications,2012,352(2):295-308.
[6] JING Z,YANG J,FENG W.Bifurcation and chaos in neural excitable system[J].Chaos Solitons&Fractals,2006,27(1):197-215.
[7] KUZNETSOV N V,LEONOV G A,VAGAITSEV V I.Analytical-numerical method for attractor localization of generalized chua's system[J].Ifac Proceedings Volumes,2010,43(11):29-33.
[8] LEONOV G A,VAGAITSEV V I,KUZNETSOV N V.Algorithm for localizing Chua attractors based on the harmonic linearization method[J].Doklady Mathematics,2010,82(1):663-666.
[9] LEONOV G A,KUZNETSOV N V,KUZNETSOVA O A,et al.Hidden oscillations in dynamical systems[J].Wseas Transactions On Systems&Control,2011,6(2):54-67.
[10] LEONOV G A,KUZNETSOV N V.Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems[J].Ifac Proceedings Volumes,2011,44(1):2494-2505.
[11] LEONOV G A,KUZNETSOV N V,VAGAITSEV V I.Localization of hidden Chua attractors[J].Physics Letters A,2011,375(23):2230-2233.
[12] LEONOV G A,KUZNETSOV N V,VAGAITSEV V I.Hidden attractor in smooth Chua systems[J].Physica D Nonlinear Phenomena,2012,241(18):1482-1486.
[13] KUZNETSOV N V,KUZNETSOVA O A,LEONOV G A,et al.Hidden attractor in Chua's circuits[C].8th International Conference on Informatics in Control,Automation and Robotics.2011:279-283.
[14] YASSEN M T.Adaptive control and synchronization of a modified Chua's circuit system[J].Applied Mathematics&Computation,2003,135(1):113-128.
[15] LEONOV G A,KUZNETSOV N V.Hidden oscillations in dynamical systems.16hilbert′s problem,Aizerman′s and Kalman′s conjectures,hidden attractors in Chua′s circuits[J].Journal Of Mathematical Sciences,2014,201(5):645-662.
[16] KUZNETSOV N V,LEONOV G A.Hidden attractors in dynamical systems:systems with no equilibria,multistability and coexisting attractors[J].IFAC Proceedings Volumes,2014,47(3):5445-5454.
[17] LEONOV G A,KISELEVA M A,KUZNETSOV N V,et al.Discontinuous differential equations:comparison of solution definitions and localization of hidden Chua attractors[J].Ifac Papersonline,2015,48(11):408-413.
[18] DUDKOWAKI D,JAFARI S,KAPITTANIAK T,et al.Hidden attractors in dynamical systems[J].Physics Reports,2016,637:1-50.
[19] LEONOV G A,KUZNETSOV N V.Hidden attractors in dynamical systems from hidden oscillations in Hilbert-Holmogorov,Aizerman,and Kalman problems to hidden chaotic attractor in Chua circuits[J].International Journal Of Bifurcation&Chaos,2013,23:1330002.
[20] TESI A,VICINO A.Robust absolute stability of lur'e control systems in parameter space[M].Pergamon Press,Inc.1991,27(1):147-151.
[21] WOLF A,SWIFT J B,SWINNEY H L,et al.Determining Lyapunov exponents from a time series[J].Physica D Nonlinear Phenomena,1985,16(3):285-317.