The coexistence of chaotic synchronization with three different nonautonomous systems under constraint conditions

Because of the inherent randomness and extreme sensitivity to initial values of chaotic systems, chaos synchronization has become the key technology for chaotic applications. With the further research, scholars have found that using continuous systems to simplify discontinuous systems cannot get accurate results, and it needs to use discontinuous models to provide a precise description. Therefore, the synchronization with three different chaotic systems will be studied by using discontinuous dynamical system theory in this paper. Taking Holmes–Duffing system as master system, Pendulum and the Stiffening Spring as slave systems, the motion laws of the three systems in different motion spaces will be discussed systematically with controllers, and the analysis conditions for synchronization to start and disappear will be given. Through numerical simulations, partial synchronization and full synchronization among three systems will be realized, which proves the validity of this method and is conducive to the further study of the synchronization of complex periodic and chaotic motions.

[1]  Giuseppe Grassi,et al.  Projective Synchronization via a Linear Observer: Application to Time-Delay, Continuous-Time and Discrete-Time Systems , 2007, Int. J. Bifurc. Chaos.

[2]  Albert C. J. Luo,et al.  Complex Dynamics of Projective Synchronization of Chua Circuits with Different Scrolls , 2015, Int. J. Bifurc. Chaos.

[3]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[4]  Amr Elsonbaty,et al.  Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system , 2016 .

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

[6]  A. El-Sayed,et al.  Analytical study of global bifurcations, stabilization and chaos synchronization of jerk system with multiple attractors , 2017 .

[7]  M. M. Al-sawalha,et al.  On Λ −ϕ generalized synchronization of chaotic dynamical systems in continuous–time , 2016 .

[8]  Fuhong Min,et al.  Synchronization of a periodically forced Duffing oscillator with a periodically excited pendulum , 2011 .

[9]  Yen-Sheng Chen,et al.  Adaptive synchronization of unidirectional and mutual coupled chaotic systems , 2005 .

[10]  Yu Feng,et al.  Switched generalized function projective synchronization of two hyperchaotic systems with hidden attractors , 2015 .

[11]  Fuhong Min,et al.  Synchronization dynamics of two different dynamical systems , 2011 .

[12]  Tao Liu,et al.  Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters , 2016 .

[13]  Yuanwei Jing,et al.  Modified projective synchronization of chaotic systems with disturbances via active sliding mode control , 2010 .

[14]  Albert C. J. Luo,et al.  Discontinuous Dynamical Systems on Time-varying Domains , 2009 .

[15]  Albert C. J. Luo,et al.  A theory for flow switchability in discontinuous dynamical systems , 2008 .

[16]  Muhammad Rehan,et al.  A concept of coupled chaotic synchronous observers for nonlinear and adaptive observers-based chaos synchronization , 2016 .

[17]  Albert C. J. Luo,et al.  A theory for synchronization of dynamical systems , 2009 .

[18]  Fuhong Min,et al.  Sinusoidal synchronization of a Duffing oscillator with a chaotic pendulum , 2011 .

[19]  Juan Luis García Guirao,et al.  New trends in nonlinear dynamics and chaoticity , 2016 .