Suppressing chaos in discontinuous systems of fractional order by active control

Abstract

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.