Abstract
Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called “self-excited attractors” has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called “hidden attractors”, which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system.
Highlights
The study of discrete-time chaotic systems has featured prominently in the past few years [1].The dynamics of a lot of chaotic systems/maps have been deeply analysed, especially with regard to the logistic map, the Arnold’s cat map, the Hénon map, the Lozi map, and the tent map [1,2]
Since the initial conditions in these systems can be only found via massive numerical search, these chaotic attractors are hard to be found [6,7]
Some research workers have recently concentrated their considerations on studying the chaotic behaviours associated with the dynamics of the Fractional-order Discrete Systems (FoDSs), i.e., maps outlined by Fractional-order Difference Equations (FoDEs) [11,12]
Summary
The study of discrete-time chaotic systems has featured prominently in the past few years [1]. Some research workers have recently concentrated their considerations on studying the chaotic behaviours associated with the dynamics of the Fractional-order Discrete Systems (FoDSs), i.e., maps outlined by Fractional-order Difference Equations (FoDEs) [11,12]. A new variable-order fractional chaotic system (defined via a piecewise constant function) has been conceived in [22] Note that all these FoDSs exhibit the “self-excited attractors”, since all initial conditions for generating chaos are placed close to the unstable fixed points [13,14,15,16].
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