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Synchronization of chaotic fractional Chen system. (English) Zbl 1080.34537

The complete synchronization of two coupled Chen’s systems with fractional derivatives is studied. The driven Chen’s system has the form

d q 1 x m dt q 1 =a(y m -x m ),
d q 2 y m dt q 2 =(c-a)x m -x m z m +cy m ,
d q 3 z m dt q 3 =x m y m -bz m ,

the response system reads

d q 1 x s dt q 1 =a(y s -x s ),
d q 2 y s dt q 2 =(c-a)x s -x m z s +cy s +u(y s -y m ),
d q 3 z s dt q 3 =x m y s -bz s ,

where u is a control parameter, (x m ,y m ,z m ) and (x s ,y s ,z s ) are phase variables for the drive and response systems, respectively. d q i /dt q i ,i=1,2,3, are the fractional derivatives with q 1 =0·86, q 2 =0·88, and q 3 =0·86.

Using Laplace transform theory, the authors provide conditions for synchronization. The technique given in the paper can be used to study synchronization of other systems with fractional derivatives.

MSC:
34D05Asymptotic stability of ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
26A33Fractional derivatives and integrals (real functions)