Projective synchronization in autonomous chaotic system via tracking control 1 by ProQuest


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									    Scientia Magna
  Vol. 5 (2009), No. 4, 103-108

      Projective synchronization in autonomous
        chaotic system via tracking control 1
                            Lixin Yang, Wansheng He and Xiaojun Liu

            Department of Mathematics and Statistics, Tianshui Normal University,
                            Tianshui, Gansu 741001, P.R.China

     Abstract This paper presents the projective synchronization of chaos systems by designing
     tracking controller based on Lyapunov stability theory. Frist, this method is implemented in
     synchronization of a simple system, then we realize the synchronization of Lu hyper-chaotic
     system. Numerical simulations show the united synchronization method works well.
     Keywords Projective synchronization, tracking control, chaotic system, Hyper-chaotic

§1. Introduction

     Synchronization is a fundamental phenomenon that enables coherent behavior in coupled
systems. In 1990, pecora and carroll proposed a successful method to synchronize two identical
chaotic systems with different initial conditions [1]. Chaos synchronization has received a
significant attention in the last few years due to its potential applications [3-14]. There exist
many types of synchronization such as complete synchronization [2], anti-synchronization [4].
Mainieri and Rehacek [10] reported a new form of chaos synchronization, termed as projective
synchronization, that the drive and response systems could be synchronized up to a scaling
factor (a proportional relation), which is usually observable in a class of systems with partial
linearity. In this regard, this paper we put forward tracking control method to achieve the
projective synchronizaiton for chaotic systems. We prove the feasibility of the method from
theoretic analysis on the basis of two chaotic systems. Numerical simulation are used to verify
the effectiveness of the proposed scheme.
     We organized this paper as follows. In section 2 we discuss the design of tracking controller.
In section 3, we present an application of this approach to control of the system and numerical
simulations demonstrate the effectiveness of the proposed synchronization scheme. Finally
concluding remark and references close the paper.

  1 This work is supported by the Gansu Provincial Education Department Foundation 0808-04 and Scientific

Research Foundations of Tianshui Normal University of China TSA0938.
104                             Lixin Yang, Wansheng He and Xiaojun Liu                       No. 4

§2. Design of controller
      Consider nonlinear chaotic system as follows
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