Abstract
This paper presents three important phenomena of chaos synchronization between two different hyperchaotic systems using nonlinear adaptive control strategy. In detailed, complete synchronization, anti- synchronization 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.
Highlights
Various types of synchronization phenomena have been presented such as complete synchronization (CS), generalized synchronization (GS), lag synchronization, antisynchronization (AS), projective synchronization (PS), generalized projective synchronization (GPS)
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
Summary
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]. The most familiar synchronization phenomena are complete synchronization and anti- synchronized [14], [15]. 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. 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]. 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
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