Abstract
Fractional calculus in discrete-time systems is a recent research topic. The fractional maps introduced in the literature often display chaotic attractors belonging to the class of “self-excited attractors”. The field of fractional map with “hidden attractors” is completely unexplored. Based on these considerations, this paper presents the first example of fractional map without equilibria showing a number of hidden attractors for different values of the fractional order. The presence of the chaotic hidden attractors is validated via the computation of bifurcation diagrams, maximum Lyapunov exponent, 0–1 test, phase diagrams, complexity, and entropy. Finally, an active controller with the aim for stabilizing the proposed fractional map is successfully designed.
Highlights
Continuous-time and discrete-time chaotic dynamical systems have been extensively studied over the last few years [1], and a large number of papers on chaos control, synchronization, and chaos application has been introduced continuously [2,3]
This paper presents a fractional map without equilibria on the basis of reference [16], which shows a number of hidden attractors for different values of the fractional order in the difference equations
Since the field of fractional map showing “hidden attractors” is completely unexplored in the literature, this paper has made a contribution to the topic by presenting the first example of fractional map without equilibria
Summary
Continuous-time and discrete-time chaotic dynamical systems have been extensively studied over the last few years [1], and a large number of papers on chaos control, synchronization, and chaos application has been introduced continuously [2,3]. Electronics 2020, 9, 748 points—for example, in systems characterized by the absence of equilibria [10] or by the presence of stable equilibria only [11] These types of attractors are the so-called “hidden attractors”, for which the initial conditions can only be found via extensive numerical search [12]. To the best of the authors’ knowledge, no fractional map showing “hidden attractors” has been introduced in the literature to date Based on these considerations, this paper presents a fractional map without equilibria on the basis of reference [16], which shows a number of hidden attractors for different values of the fractional order in the difference equations.
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