Synchronization and Antisynchronization for a Class of Chaotic Systems by a Simple Adaptive Controller

Abstract

This paper investigates the synchronization and antisynchronization for a class of chaotic system. Firstly, a necessary and sufficient condition is proposed to synchronize and antisynchronization simultaneously for the chaotic systems. Secondly, two methods are obtained to realize coexistence of synchronization and antisynchronization in the chaotic systems, and the corresponding adaptive controllers are also given. Finally, two numerical examples with simulation verify the correctness and effectiveness of the obtained results.

Hybrid Synchronization Problem of a Class of Chaotic Systems by an Universal Control Method

“…In a word, the anti-synchronization problem for a given chaotic system should be investigated firstly when this system is considered. In fact, we have proved rigorously the fact: for a given chaotic system in this form:ẋ = f (x), where x ∈ R n , the anti-synchronization problem exists if and only if f (−x) = − f (x) in [26]. Therefore, there are still exist many important problems for the chaotic systems in arbitrary dimensions which need to be investigated further.…”

Hybrid Synchronization of Uncertain Generalized Lorenz System by Adaptive Control

“…Moreover, uncertainties exist widely in engineering and they often bring adverse effects to the stability and performance of real systems. So there is an increasing demand on developing better control techniques [12][13][14][15][16].…”

A Simple Hybrid Synchronization for a Class of Chaotic Financial Systems

“…Scholars have also done a lot of related work in the financial system, as well as in the control and synchronization research [8][9][10]. So far, a lot of methods about chaotic synchronization have been presented to prove that the chaotic synchronization method feasibly [11][12][13][14][15], such as driving response synchronization, coupled synchronization, feedback synchronization, impulsive synchronization, and adaptive synchronization, for the study of chaos synchronization problem has been more mature. However, compared with the chaotic synchronization problem, the problem of antisynchronization of chaotic systems is poorly known, so is the hybrid synchronization, and there are few studies in the financial system.…”

“…In a broad sense, the synchronization and antisynchronization of chaotic systems are the special cases of the hybrid synchronization of the chaotic financial systems; that is, all the variables are synchronized or are antisynchronous [16]. In recent years, some scholars have obtained some research results in the antisynchronization of [12][13][14][15], but they are still in the initial stage; many theories are not mature yet. For example, in the study of antisynchronization problems, the equilibrium point of the error systeṁ= ( ) + ( ) is not explicitly stated, if and only if the condition of (− ) = − ( ) is satisfied [13,14], so that the study of the antisynchronization has a lot of research prospects.…”

“…And so far, in the study of chaos hybrid synchronization problem, there has not been any result of the explicit conditions or algorithms to choose which variables in the driving chaotic system can be synchronized or antisynchronized with the implementation of the corresponding variables in the response chaotic system. Scholars have made some achievements in the chaotic system in hybrid synchronization [17,18], but the hybrid synchronization of the chaotic financial systems is still not perfect so far [15][16][17]. There is still a long wall to go.…”