A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors*

Abstract

We study a novel class of two-dimensional maps with infinitely many coexisting attractors. Firstly, the mathematical model of these maps is formulated by introducing a sinusoidal function. The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable. In particular, a computer searching program is employed to explore the chaotic attractors in these maps, and a simple map is exemplified to show their complex dynamics. Interestingly, this map contains infinitely many coexisting attractors which has been rarely reported in the literature. Further studies on these coexisting attractors are carried out by investigating their time histories, phase trajectories, basins of attraction, Lyapunov exponents spectrum, and Lyapunov (Kaplan–Yorke) dimension. Bifurcation analysis reveals that the map has periodic and chaotic solutions, and more importantly, exhibits extreme multi-stability.