Three-dimensional Chaotic Fractional Maps without Fixed Points: Dynamics, Coexisting Hidden Attractors and Hardware Implementation

Abstract

In recent years there has been a great interest in investigating the dynamic properties of fractional systems, described by differential or difference equations of non-integer order. A number of manuscripts have been written regarding the presence of chaos in fractional maps characterized by the existence of both “self-excited attractors” and “hidden attractors However, most of the papers have dealt with the underlying theory in chaotic dynamics, rather than the hardware implementation of the maps. Specifically, referring to the realization of fractional maps with “hidden attractors no paper has been published so far. The paper aims to bridge the gap between theory and implementation by presenting the first example of hardware realization of a three-dimensional (3D) fractional map characterized by coexisting chaotic hidden attractors. The manuscript exploits first the Grunwald-Letnikov fractional difference operator to introduce a new class of chaotic discrete systems without fixed points. Then, the coexistence of several chaotic hidden attractors is shown, along with the coexistence of a number of bifurcations, depending on the values of the initial conditions. Finally, the conceived 3D fractional map is implemented in hardware via a microcontroller. The motivation is to probe the presence of coexisting chaotic hidden attractors in real working systems described by non-integer order difference equations.