Combination complex synchronization of three chaotic complex systems

In this paper, we firstly design a chaotic complex system and study its dynamical properties including invariance, dissipativity, equilibria, Lyapunov exponents, chaotic behavior, as well as chaotic attractors. What is more, the scaling matrices are always chosen as real matrices in previous combination synchronization schemes within two drive real systems and one response real system evolving in the same or inverse directions simultaneously. However, in many real-life applications, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, combination synchronization with regard to the complex scaling matrices, referred as combination complex synchronization, will be made the further research about three chaotic complex systems. Based on Lyapunov stability theory, three identical chaotic complex systems are considered and the corresponding controllers are designed to achieve the complex combination synchronization. The corresponding theoretical proofs and numerical simulations are given to demonstrate the validity and feasibility of the presented control technique.

[1]  Junwei Sun,et al.  Modified projective and modified function projective synchronization of a class of real nonlinear systems and a class of complex nonlinear systems , 2014 .

[2]  Gamal M. Mahmoud,et al.  Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system , 2007 .

[3]  Luo Runzi,et al.  Combination synchronization of three classic chaotic systems using active backstepping design. , 2011, Chaos.

[4]  Guangzhao Cui,et al.  Combination–combination synchronization among four identical or different chaotic systems , 2013 .

[5]  Mark J. McGuinness,et al.  The complex Lorenz equations , 1982 .

[6]  Emad E. Mahmoud,et al.  On the hyperchaotic complex Lü system , 2009 .

[7]  Xingyuan Wang,et al.  Chaos in the fractional-order complex Lorenz system and its synchronization , 2013 .

[8]  Quan Yin,et al.  Compound synchronization for four chaotic systems of integer order and fractional order , 2014 .

[9]  Leon O. Chua,et al.  Simplest Chaotic Circuit , 2010, Int. J. Bifurc. Chaos.

[10]  Emad E. Mahmoud,et al.  Complete synchronization of chaotic complex nonlinear systems with uncertain parameters , 2010 .

[11]  Hao Zhang,et al.  Module-phase synchronization in hyperchaotic complex Lorenz system after modified complex projection , 2014, Appl. Math. Comput..

[12]  Tassos Bountis,et al.  Active Control and Global Synchronization of the Complex Chen and lÜ Systems , 2007, Int. J. Bifurc. Chaos.

[13]  Xiaomei Yan,et al.  Modified projective synchronization of fractional-order chaotic systems based on active sliding mode control , 2013, 2013 25th Chinese Control and Decision Conference (CCDC).

[14]  Stephane Pernot,et al.  Design criteria for optimally tuned nonlinear energy sinks—part 1: transient regime , 2012 .

[15]  Junwei Sun,et al.  Quasi-Ideal Memory System , 2015, IEEE Transactions on Cybernetics.

[16]  Xing-yuan Wang,et al.  Projective synchronization of fractional order chaotic system based on linear separation , 2008 .

[17]  Manfeng Hu,et al.  Hybrid projective synchronization in a chaotic complex nonlinear system , 2008, Math. Comput. Simul..

[18]  Xing-yuan Wang,et al.  Adaptive control for synchronization of a four-dimensional chaotic system via a single variable , 2011 .

[19]  Emad E. Mahmoud,et al.  Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems , 2010 .

[20]  Chao Luo,et al.  Hybrid modified function projective synchronization of two different dimensional complex nonlinear systems with parameters identification , 2013, J. Frankl. Inst..

[21]  Jie Chen,et al.  Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control , 2014 .

[22]  Yi Shen,et al.  Compound synchronization of four memristor chaotic oscillator systems and secure communication. , 2013, Chaos.

[23]  Ping Liu,et al.  Adaptive anti-synchronization of chaotic complex nonlinear systems with unknown parameters , 2011 .

[24]  Junwei Sun,et al.  Transmission projective synchronization of multi-systems with non-delayed and delayed coupling via impulsive control. , 2012, Chaos.

[25]  Da Lin,et al.  Module-phase synchronization in complex dynamic system , 2010, Appl. Math. Comput..