A Discrete Map with a Hyperbolic Sine Function: Dynamics, Stabilization, and Synchronization

In this paper, a fractional-order discrete map with a hyperbolic sine function has been proposed and studied. Firstly, the basic characteristics of the map in integer-order case are studied theoretically and numerically. Secondly, dynamics of the map are investigated via numerical simulations. Attractors and bifurcation diagram spectrums are given when a parameter is varied. Furthermore, the map with the Caputo fractional difference operator has been studied. The chaotic attractors in commensurate-order and incommensurate-order cases are shown. For the characteristics of hyperbolic sine function, the chaotic attractors with different structures for the map can be obtained. It can be concluded that the map has rich dynamics in integer-order and fractional-order cases. Finally, stabilization and adaptive synchronization of the fractional-order map are realized by designing suitable controllers, respectively. Numerical results are used to demonstrate the effectiveness of the controllers for the map.