Suppressing chaos in discontinuous systems of fractional order by active control

- Chaotic piecewise continuous systems of fractional order are investigated.
- Regularization by differential inclusion is applied and hence a continuous approximation by the Cellina's Theorem is used.
- Stability of piecewise continuous systems of fractional order is analyzed.
- An active control technique, based on stabilization of unstable equilibria, is proposed and implemented for chaos control.
- Numerical simulations are presented for the fractional Shimizu-Morioka's system.

In this paper, a chaos control algorithm for a class of piece-wise continuous chaotic systems of fractional order, in the Caputo sense, is proposed. With the aid of Filippov's convex regularization and via differential inclusions, the underlying discontinuous initial value problem is first recast in terms of a set-valued problem and hence it is continuously approximated by using Cellina's Theorem for differential inclusions. For chaos control, an active control technique is implemented so that the unstable equilibria become stable. As example, Shimizu-Morioka's system is considered. Numerical simulations are obtained by means of the Adams-Bashforth-Moulton method for differential equations of fractional-order.