Hybrid synchronization of n–scroll Chua and Lur’e chaotic systems via backstepping control with novel feedback

This paper investigates the backstepping control design with novel feedback input approach for controlling chaotic systems to guarantee the complete synchronization as well as the anti-synchronization of chaotic systems, viz. n–scroll Chua (K. Wallace et.al. 2001) and Lur’e chaotic systems. Our theorems on hybrid synchronization for n–scroll Chua and Lur’e (J.Suyken et.al. 1997) chaotic systems is established using Lyapunov stability theory. Based on the Lyapunov function, the backstepping control is determined to tune the controller gain based on the precalculated feedback control inputs. The backstepping scheme is recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global stability performance of strict-feedback chaotic systems. Since the Lyapunov exponents are not required for these calculations, the backstepping control method is effective and convenient to synchronize the chaotic systems. Mainly this technique gives the flexibility to construct a control law. Numerical simulations are also given to illustrate and validate the hybrid synchronization results derived in this paper.

[1]  Lixin Tian,et al.  Global synchronization for time-delay of WINDMI System , 2006 .

[2]  L. Chua,et al.  Generalized synchronization of chaos via linear transformations , 1999 .

[3]  Bernd Blasius,et al.  Complex dynamics and phase synchronization in spatially extended ecological systems , 1999, Nature.

[4]  Jiang Wang,et al.  Unidirectional synchronization for Hindmarsh-Rose neurons via robust adaptive sliding mode control , 2010 .

[5]  S. M. Lee,et al.  Adaptive synchronization of Genesio-Tesi chaotic system via a novel feedback control , 2007 .

[6]  Bin Deng,et al.  Synchronization of Ghostburster neuron in external electrical stimulation via H∞ variable universe fuzzy adaptive control , 2009 .

[7]  Jian-ping Yan,et al.  Generalized projective synchronization for the chaotic Lorenz system and the chaotic Chen system , 2006 .

[8]  J. Suykens,et al.  Nonlinear H∞ Synchronization of Chaotic Lur'e Systems , 1997 .

[9]  Suochun Zhang,et al.  Adaptive backstepping synchronization of uncertain chaotic system , 2004 .

[10]  Ju H. Park.  Synchronization of Genesio chaotic system via backstepping approach , 2006 .

[11]  Sundarapandian Vaidyanathan,et al.  Generalized Projective Synchronization of Double-Scroll Chaotic Systems Using Active Feedback Control , 2012 .

[12]  J. Yorke,et al.  Chaos: An Introduction to Dynamical Systems , 1997 .

[13]  Jun-an Lu,et al.  Adaptive feedback synchronization of a unified chaotic system , 2004 .

[14]  A. Fuller,et al.  Stability of Motion , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[15]  Jun-an Lu,et al.  Parameter identification and backstepping control of uncertain Lü system , 2003 .

[16]  J. Suykens,et al.  Robust synthesis for master-slave synchronization of Lur'e systems , 1999 .

[17]  Guanrong Chen,et al.  Generation of n-scroll attractors via sine function , 2001 .

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

[19]  Wei Xu,et al.  Adaptive generalized projective synchronization in different chaotic systems based on parameter identification , 2007 .

[20]  Ju H. Park.  Letter to the Editor: Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters , 2008 .

[21]  Z. Ge,et al.  Phase synchronization of coupled chaotic multiple time scales systems , 2004 .

[22]  J. Hindmarsh,et al.  A model of neuronal bursting using three coupled first order differential equations , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[23]  M. Lakshmanan,et al.  SECURE COMMUNICATION USING A COMPOUND SIGNAL USING SAMPLED-DATA FEEDBACK , 2003 .

[24]  Sundarapandian Vaidyanathan,et al.  Generalized Projective Synchronization of Three-Scroll Chaotic Systems via Active Control , 2012 .

[25]  Rui-hong Li,et al.  A special full-state hybrid projective synchronization in symmetrical chaotic systems , 2008, Appl. Math. Comput..

[26]  Ljupco Kocarev,et al.  General approach for chaotic synchronization with applications to communication. , 1995, Physical review letters.

[27]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems , 1983 .

[28]  F. M. Kakmeni,et al.  CHAOS SYNCHRONIZATION IN BI-AXIAL MAGNETS MODELED BY BLOCH EQUATION , 2006 .

[29]  V. Sundarapandian,et al.  Global Chaos Synchronization of the Pehlivan Systems by Sliding Mode Control , 2011 .

[30]  Ju H. Park,et al.  A novel criterion for delayed feedback control of time-delay chaotic systems , 2005 .

[31]  Z. Guan,et al.  Generalized synchronization of continuous chaotic system , 2006 .

[32]  Parlitz,et al.  General approach for chaotic synchronization with applications to communication. , 1995, Physical Review Letters.

[33]  Y. Kuramoto,et al.  Dephasing and bursting in coupled neural oscillators. , 1995, Physical review letters.

[34]  Q. Jia.  Projective synchronization of a new hyperchaotic Lorenz system , 2007 .

[35]  Louis M. Pecora,et al.  Synchronizing chaotic circuits , 1991 .

[36]  Sundarapandian Vaidyanathan,et al.  New Results on the Global Chaos Synchronization for Liu-Chen-Liu and Lü Chaotic Systems , 2010 .

[37]  Lansun Chen,et al.  Complexity of an SIR epidemic dynamics model with impulsive vaccination control , 2005 .

[38]  H. Yau.  Design of adaptive sliding mode controller for chaos synchronization with uncertainties , 2004 .

[39]  Zuolei Wang.  Chaos synchronization of an energy resource system based on linear control , 2010 .

[40]  M. Lakshmanan,et al.  Chaos in Nonlinear Oscillators: Controlling and Synchronization , 1996 .

[41]  Sundarapandian Vaidyanathan,et al.  Hybrid Synchronization of Arneodo and Rössler Chaotic Systems by Active Nonlinear Control , 2012 .