Bifurcations, Hidden Chaos and Control in Fractional Maps
Adel Ouannas, Othman Abdullah Almatroud, Amina–Aicha Khennaoui, M. Mossa Al-Sawalha, Dumitru Bǎleanu, Van Van Huynh, Viet–Thanh Pham
Abstract
Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. Results show that the dynamics of the proposed fractional maps are sensitive to both initial conditions and fractional order. There are coexisting attractors which have been displayed in terms of bifurcation diagrams, phase portraits and a 0-1 test. Furthermore, control schemes are introduced to stabilize the chaotic trajectories of the two novel systems.