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.
Highlights
Continuous-time and discrete-time chaotic dynamical systems have been extensively studied over the last years
Fractional maps with hidden attractors have rarely been reported [23], which has inspired researchers to devote themselves to the design of new two and three-dimensional fractional discrete-time chaotic systems [24,25,26]
We propose two active controllers with the aim of stabilizing the chaotic dynamics of the two fractional maps
Summary
Continuous-time and discrete-time chaotic dynamical systems have been extensively studied over the last years. Referring to discrete-time systems number of chaotic maps have been deeply analyzed Researchers such as Hénon, Lozi and Arnold have attempted to provide different maps with different features. Researchers have extensively examined the potential application of these maps in many fields such as, engineering, economics and other areas [12,13,14] For this purpose, many fractional maps have been reported in the literature to show the different dynamical phenomena. Fractional maps with hidden attractors have rarely been reported [23], which has inspired researchers to devote themselves to the design of new two and three-dimensional fractional discrete-time chaotic systems [24,25,26]. We propose two active controllers with the aim of stabilizing the chaotic dynamics of the two fractional maps
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