Three Important Phenomena of Chaos Synchronization Between Two Different Hyperchaotic Systems via Adaptive Control Method

This paper presents three important phenomena of chaos synchronization between two different hyperchaotic systems using nonlinear adaptive control strategy. In detailed, complete synchronization, antisynchronization and hybrid synchronization with nine unknown parameters. Modified hyperchaotic Pan system is consider as drive and hyperchaotic Liu system as response system. Stabilization of error dynamics for each phenomenon is realized by satisfying Lyapunov's second method as a main tool. Theoretical analysis and numerical simulations are shown to verify the results.


Introduction
Since the pioneering work of Pecora and Carroll in 1990s, on the synchronization of chaotic systems, synchronization phenomenon has formed a new body of research activities which is at the fore front of recent application topic in nonlinear dynamics [1][2][3][4][5][6], and opens the way for chaotic systems synchronization and a various techniques such as adaptive control, active control, nonlinear control, sliding mod control, back-stepping design method and so on have been successfully applied to chaos control and synchronization [7][8][9][10][11][12][13].
Complete synchronization is characterized by the equality of state variables evolving in the time, while antisynchronization is characterized by the disappearance of sum of relevant variables evolving in the time. Projective synchronization is characterized by the fact that the drive and response systems could be synchronized up to a scaling factor, whereas in generalized projective synchronization, the responses of the synchronized dynamical states synchronize up to a constant scaling matrix . It is easy to see that the complete synchronization and antisynchronization are special cases of generalized projective synchronization where the scaling matrix = and = − , respectively [16].
In the hybrid synchronization scheme, one part of the system is synchronized and the other part is antisynchronized so that the complete synchronization and antisynchronization coexist in the system [17].
In this paper, we discuss some important phenomena for chaos synchronization i.e. complete synchronization, antisynchronization and hybrid synchronization between two different 4D hyperchaotic systems via adaptive control, The results derived in this paper are established using the Lyapunov's second method.

Description of Hyperchaotic Modified Pan and Liu Systems
According to the Ref [18], the modified hyperchaotic Pan system which described by the following dynamical system where , , , are the state variables and , , , 0 are the parameters of the system. When parameters = 10, = 8 3 ⁄ , = 28 and = 10, system (1) is hyperchaotic and has two positive Lyapunov exponents, i.e. = 0.24784, = 0.08194, and has only one equilibrium " 0,0,0,0 . This equilibrium is an unstable under these parameters.
In 2006, Wang et al. [19], presented the four-dimensional autonomous Liu system which described by where a, r, k, h, p, q 0 are system parameters. When parameters a = 10, p = 2.5, r = 40, q = 10.6, k = 1 and h = 4, system (2) is hyperchaotic and has only equilibrium O 0,0,0,0 , and the equilibrium is an unstable saddle node under these parameters. Fig. 1 and Fig. 2 shows the attractors of the system (1) and the system (2) respectively.

Complete Synchronization Between Modified Pan and Liu Systems
In this section, the synchronization behavior between two different hyperchaotic systems with nine unknown parameters is achieved based on the Lyapunov second method and adaptive control strategy. In order to observe complete synchronization between hyperchaotic Modified Pan system and hyperchaotic Liu system, we consider the system (1) as the drive system and hyperchaotic Liu system as the response system which describe by the following system where 5 = 65 , 5 , 5 , 5 7 8 is the controller to be designed. The synchronization error 9:; is defined as: where < is a scaling factor taken the value 1 for complete synchronization and -1 for anti-synchronization according to the projective synchronization approach. So, subtracting the above system from the system (1), we get the error dynamical system between the drive system and the response system which is given by: # $ % 9 = 9 − 9 + 5 9 = &9 + 9 + & − − '4 4 + + 5 9 = −)9 + − ) + ℎ4 − + 5 9 = − *9 − 9 − * + 4 + 5 We need to find the nonlinear adaptive control law for 5 < , ∀> and parameters estimation rules to guarantee that the error dynamics of (5) is globally asymptotically stable.
Hence, we arrive at the following results. Theorem 1. The non-identical system (1) and system (3) with nine unknown parameters are complete synchronize by nonlinear adaptive control law (6), and the update law for parameters is chosen as (13).

Hybrid Synchronization Between Modified Pan and Liu Systems
Based on the adaptive control and Lyapunov second method the hybrid synchronization between two different 4D hyperchaotic systems, i.e. Modified Pan and Liu systems is consider in this section. Modified hyperchaotic Pan system is taken as drive (system 1) and hyperchaotic Liu system as response (system 3).
Therefore, hybrid synchronization between two hyperchaotic systems (1) and (3) with nine unknown parameters is achieved. The proof is now complete based on Lyapunov method. Fig. 7 and Fig. 8 show verify these results numerically.

Conclusions
In this paper deals with chaos synchronization between