Smooth stable manifolds for the non-instantaneous impulsive equations with applications to Duffing oscillators

Abstract

In this paper, we present a theory of smooth stable manifold for the non-instantaneous impulsive differential equations on the Banach space or Hilbert space. Assume that the non-instantaneous linear impulsive evolution differential equation admits a uniform exponential dichotomy, we give the conditions of the existence of the global and local stable manifolds. Furthermore, $C^k$ -smoothness of the stable manifold is obtained, and the periodicity of the stable manifold is given. Finally, an application to nonlinear Duffing oscillators with non-instantaneous impulsive effects is given, to demonstrate the existence of stable manifold.

Keywords: exponential dichotomy; impulsive; non-instantaneous; smooth stable manifold.