A scheme for designing extreme multistable discrete dynamical systems

Abstract

In this paper, we propose a scheme for designing discrete extreme multistable systems coupling two identical dynamical systems. Existence of infinitely many attractors in the system is obtained via partial synchronization between two systems for a given set of parameters. We give a conjecture that extreme multistable systems can be designed by coupling two m-dimensional dynamical systems in such a way that i \((1 \le i\le m-1)\) number of state variables of the two systems synchronize completely and \((m-i)\) number of state variables keep constant difference. We demonstrate the applicability of our scheme in two-dimensional (2D) as well as three-dimensional (3D) discrete dynamical systems. In particular, we discuss our scheme taking coupled 2D Henon maps, coupled 2D Duffing maps and coupled 3D Henon maps. We have analytically shown the existence of fixed points and period-2 orbits in the coupled system with the variation of initial conditions. These analytically derived conditions matched very well with the numerical simulation results. Variation of the largest Lyapunov exponent with the initial conditions is shown to confirm the existence of extreme multistability in the model. Our scheme may be useful for designing physically, chemically and biologically useful multistable discrete dynamical systems.