## The dynamics and synchronization of a fractional-order system with complex variables.(English)Zbl 1472.34102

Summary: A fractional-order system with complex variables is proposed. Firstly, the dynamics of the system including symmetry, equilibrium points, chaotic attractors, and bifurcations with variation of system parameters and derivative order are studied. The routes leading to chaos including the period-doubling and tangent bifurcations are obtained. Then, based on the stability theory of fractional-order systems, the scheme of synchronization for the fractional-order complex system is presented. By designing appropriate controllers, the synchronization for the system is realized. Numerical simulations are carried out to demonstrate the effectiveness of the proposed scheme.

### MSC:

 34D06 Synchronization of solutions to ordinary differential equations 34A08 Fractional ordinary differential equations 34H05 Control problems involving ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations

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