Nonlinear dynamics of three particular two-dimensional maps under an additive constant perturbation

Abstract

Discrete dynamical systems which are described by nonlinear maps possess a rich spectrum of dynamical behaviours as chaos in many aspects. The reason for chaotic behaviour is not the complexity of a dynamical systems but the action of external perturbations. The external perturbation in discrete dynamical systems alters the system’s dynamics to a great extent. In this paper, we numerically analyze the local dynamics and bifurcation of three particular two-dimensional maps such as Burger’s map, Tingerbell map and Predator-Prey map under an additive constant perturba-tion. Due to the external perturbation, the systems show more complex dynamical behaviours than that of the original system. The basic properties of the dynamics are analyzed by bifurcation diagram, phase portrait and Poincaré map. With the change of control parameters values, the perturbed maps exhibit new and interesting dynamical behaviours. More specifically, this paper reports the findings of chaos, a route from an invariant circle to transient chaos with a great abundance of periodic windows including period-doubling and reverse period-doubling bifurcations, interior crises and coexisting of chaotic attractors. In addition, suppression and enhancement of chaos have been found from the range of control parameters of the system.