Bifurcation analysis and chaos control of the modified Chua’s circuit system☆

Abstract From the view of bifurcation and chaos control, the dynamics of modified Chua’s circuit system are investigated by a delayed feedback method. Firstly, the local stability of the equilibria is discussed by analyzing the distribution of the roots of associated characteristic equation. The regions of linear stability of equilibria are given. It is found that there exist Hopf bifurcation and Hopf-zero bifurcation when the delay passes though a sequence of critical values. By using the normal form method and the center manifold theory, we derive the explicit formulas for determining the direction and stability of Hopf bifurcation. Finally, chaotic oscillation is converted into a stable equilibrium or a stable periodic orbit by designing appropriate feedback strength and delay. Some numerical simulations are carried out to support the analytic results.

[1]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[2]  Tingwen Huang,et al.  A new method to control chaos in an economic system , 2010, Appl. Math. Comput..

[3]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[4]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[5]  Jürgen Kurths,et al.  Stochastic and Coherence resonances in a modified Chua's Circuit System with Multi-scroll orbits , 2013, Int. J. Bifurc. Chaos.

[6]  Yun Chen,et al.  Global anti-synchronization of master-slave chaotic modified Chua's circuits coupled by linear feedback control , 2010, Math. Comput. Model..

[7]  Olfa Boubaker,et al.  Chaos synchronization for master slave piecewise linear systems: Application to Chua's circuit , 2012 .

[8]  P. Saha,et al.  Multiple delay Rössler system—Bifurcation and chaos control , 2008 .

[9]  Tom T. Hartley,et al.  Control of Chua's Circuit , 1993, J. Circuits Syst. Comput..

[10]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[11]  Junjie Wei,et al.  Bifurcation analysis for Chen's system with delayed feedback and its application to control of chaos , 2004 .

[12]  Elena Braverman,et al.  Mackey-Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control , 2013, Appl. Math. Comput..

[13]  M. T. Yassen,et al.  Adaptive control and synchronization of a modified Chua's circuit system , 2003, Appl. Math. Comput..

[14]  K. Thamilmaran,et al.  Hyperchaos in SC-CNN based modified canonical Chua’s circuit , 2014 .

[15]  Tzuyin Wu,et al.  Chaos control of the modified Chua's circuit system , 2002 .

[16]  Ben Niu,et al.  Bifurcation analysis in the control of chaos by extended delay feedback , 2013, Journal of the Franklin Institute.

[17]  Xiaofeng Liao,et al.  Controlling chaotic Chua's circuit based on piecewise quadratic Lyapunov functions method , 2004 .

[18]  Fang Tang,et al.  An adaptive active control for the modified Chua's circuit , 2005 .

[19]  Fei Xu,et al.  Global Analysis on n-scroll Chaotic attractors of Modified Chua's Circuit , 2009, Int. J. Bifurc. Chaos.

[20]  Pietro S. Pantano,et al.  Synchronization and Waves in a Ring of Diffusively Coupled Memristor-Based Chua’s Circuits , 2014 .