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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 \[ \frac{d^{q_1}x_m}{dt^{q_1}} = a(y_m-x_m), \]
\[ \frac{d^{q_2}y_m}{dt^{q_2}} = (c-a)x_m -x_m z_m +c y_m, \]
\[ \frac{d^{q_3}z_m}{dt^{q_3}} = x_m y_m -b z_m, \] the response system reads \[ \frac{d^{q_1}x_s}{dt^{q_1}} = a(y_s-x_s), \]
\[ \frac{d^{q_2}y_s}{dt^{q_2}} = (c-a)x_s -x_m z_s +c y_s + u(y_s-y_m), \]
\[ \frac{d^{q_3}z_s}{dt^{q_3}} = x_m y_s -b z_s, \] where \(u\in \mathbb{R}\) 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:
34D05 Asymptotic properties of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
26A33 Fractional derivatives and integrals
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