A Quadratic Fractional Map without Equilibria: Bifurcation, 0–1 Test, Complexity, Entropy, and Control
Adel Ouannas, Amina–Aicha Khennaoui, Shaher Momani, Giuseppe Grassi, Viet–Thanh Pham, Reyad El-Khazali, Duy Võ Hoàng
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.