Suppressing chaos in discontinuous systems of fractional order by active control

TL;DR

This paper introduces an active control method to suppress chaos in fractional-order discontinuous systems, using differential inclusions and numerical simulations to demonstrate effectiveness on the Shimizu--Morioka system.

Contribution

It presents a novel chaos control algorithm for fractional-order discontinuous systems utilizing differential inclusions and active control, with validation through numerical simulations.

Findings

Chaos is successfully suppressed in the tested system.

The control method stabilizes unstable equilibria.

Numerical results confirm the effectiveness of the approach.

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.