Global Attractors for a Class of Discrete Dynamical Systems

Abstract

In this paper, we study the existence of global attractors for a class of discrete dynamical systems naturally originated from impulsive dynamical systems. We establish sufficient conditions for the existence of a discrete global attractor. Moreover, we investigate the relationship among different types of global attractors, i.e., the attractor \({\mathcal {A}}\) of a continuous dynamical system, the attractor \(\tilde{{\mathcal {A}}}\) of an impulsive dynamical system and the attractor \(\hat{{\mathcal {A}}}\) of a discrete dynamical system. Two applications are presented, one involving an integrate-and-fire neuron model, and the other involving a nonlinear reaction-diffusion initial boundary value problem.