## Chaos, feedback control and synchronization of a fractional-order modified autonomous Van der Pol-Duffing circuit.(English)Zbl 1221.93227

Summary: Stability analysis of the fractional-order modified Autonomous Van der Pol–Duffing (MAVPD) circuit is studied using the fractional Routh–Hurwitz criteria. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than 3. Furthermore, the fractional Routh–Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh–Hurwitz conditions and using specific choice of linear controllers, it is shown that the fractional-order MAVPD system is controlled to its equilibrium points; however, its integer-order counterpart is not controlled. Moreover, chaos synchronization of MAVPD system is found only in the fractional-order case when using a specific choice of nonlinear control functions. This shows the effect of fractional order on chaos control and synchronization. Synchronization is also achieved using the unidirectional linear error feedback coupling approach. Numerical results show the effectiveness of the theoretical analysis.

### MSC:

 93D15 Stabilization of systems by feedback 34A08 Fractional ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37N35 Dynamical systems in control

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