A note on synchronization of a fractional order Coullet chaotic system

Abstract—In this paper, chaos synchronization problem of the fractional order Coullet system in a master-slave pattern is investigated by using the nonlinear feedback control method. Suitable synchronization conditions are analyzed based on the Lyapunov stability theory. And the synchronization of commensurate order Coullet chaotic system of the base order 0.98 is implemented by virtue of the method. Numerical simulations are provided to verify the performance of the proposed controller. Keyword-Chaotic synchronization, Fractional order Coullet chaotic system, master-slave, Nonlinear control, Fractional differential equations
Coullet system is proposed. In Section Ⅴ , numerical simulations are presented to demonstrate the effectiveness of the proposed method. Finally, conclusions are given in Section Ⅵ. II. IONAL ORDER
d q X1 = F(X1) , dt q
Where
(3)
X 1 ∈ ℜ n×1 , F ( X 1 ) = ( f ( x1 ), f ( x 2 ),..., f ( x n )) T
n×1
and
may be 0 < q ≤ 1 . The fractional order q ∈ ℜ unequal. The equilibrium points of system (3) can be derived by solving following equation F(X1) = 0 , (4) The stability of fractional order system has been thoroughly investigated, and necessary and sufficient conditions have been presented in Ref.[13]. Let the corresponding response(slave) system be
A note on synchronization of a fractional order Coullet chaotic system
Wen Tan 1, 2 ,Fengling Jiang 2 , Jian Xun Liu 1 ,Min Chen 2
1. Key Lab of Knowledge Processing and Networked Manufacture, Hunan University of Science and Technology, Xiangtan, Hunan, 411201,China 2. College of Information & Electrical Engineering, Hunan university of Science and Technology, Xiangtan,Hunan,411201,China wentan168@
(2)
α
Here Γ denotes Gamma function and the operator D* is called Caputo differential operator of order α . Given a fractional order chaotic system, i.e., the drive(master) system is
SYSTEM’S SYNCHRONIZATION THEORY
At present, there are many definitions of the fractional derivatives[9,10], in the paper, the following definition is used
dqX2 (5) = F ( X 2 ) + u (t ) , dt q n×1 where X 2 ∈ ℜ , u (t ) is control function. Suppose the
error between the system (3) and the system (5) is
e(t ) = X 1 (t ) − X 2 (t ) , then the fractional error system
less than α . J ( β > 0) is the β -order Riemann-Liouville integer operator which satisfies
β
J β y (t ) =
β −1 t 1 ( t − τ ) y (τ ) d τ , β > 0 , Γ ( β ) ∫0
can be obtained as
order 0 < q ≤ 1 is asymptotically stable corresponding integer order system (8) is stable.
if
the
d qe = F ( X 1 ) − F ( X 2 ) − u (t ) , dt q
α D* x(t ) = J m −α x ( m) (t ) , (α > 0) , (1) Where m = [α ] ,i.e., m is the first integer which is not
I.
INTRODUCTION
The fractional differential calculus dates from 17th century, but until recent 10 years that it was applied to physics and engineering[1,2]. It was found that many systems in interdisciplinary fields could be described by the fractional differential equations, such as viscoelastic systems,dielectric polarization, electrode-electrolyte polarization and electromagnetic waves[1]. Nowadays, it was known that the fractional order systems possess memery and display much more sophisticated dynamics compared to its integral order counterpart, which is of great significance in secure communication and control process. Furthermore, many fractional order dynamical systems have been examined to exhibit chaotic behavior[3]. Recently, Chaos synchronization problem of fractional order differential systems are being widely investigated[4-7], for different chaotic systems, various relevant control methods are used to deal with them. So far, there is only a report on control and synchronization of a fractional order Coullet chaotic system[8]. Unfortunately, the numerical simulation results presented in the above paper are not consistent with reality.Accordingly,This paper focuses on synchronization of two fractional order Coullet chaotic systems. In virtue of the Lyapunov stability theory, suitable conditions for nonlinear feedback control are derived and further verified through simulations. The rest of the paper is organized as follows. In Section Ⅱ , fractional differential and fractional order system’s synchronization theory are introduced. In Section Ⅲ, the fractional order Coullet system is described. In Section Ⅳ, the nonlinear feedback synchronization of fractional order