Abstract
In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.
Highlights
Academic Editors: KarthikeyanIn the last few years, the study of discrete chaotic systems has been a point of discussion in the fields of control and secure communication
Inspired by the aforementioned research background, this paper presents a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations
The basic properties involving the stability of the equilibrium points and symmetry of the map are studied by theoretical analysis
Summary
In the last few years, the study of discrete chaotic systems has been a point of discussion in the fields of control and secure communication. Recent reports discuss subjects including: the novel convenient condition for the stability of fractionalorder difference systems in the incommensurate-order case [30]; the complex dynamics in the discrete memristor-based system with fractional-order difference [31]; the chaos and projective synchronization of a fractional-order difference map with no equilibria [32]; and the rich dynamical characteristics of a new fractional-order, 2D discrete chaotic map [33]. These works mainly focus on the stability, dynamics, bifurcation, and synchronization of fractional-order discrete maps.
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