Linearization of Impulsive Differential Equations with Ordinary Dichotomy

Abstract

This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear system $\dot{x}(t)=A(t)x(t)+f(t,x)$, $t\ne t_k$, $\mathrm{\Delta }x(t_k)=\tilde{A}(t_k)x(t_k)+\tilde{f}(t_k,x)$, $k\in \mathbb{Z}$, is topologically conjugated to $\dot{x}(t)=A(t)x(t)$, $t\ne t_k$, $\mathrm{\Delta }x(t_k)=\tilde{A}(t_k)x(t_k)$, $k\in \mathbb{Z}$, where $\mathrm{\Delta }x(t_k)=x(t_k^{+})-x(t_k^{-})$, $x(t_k^{-})=x(t_k)$, represents the jump of the solution $x(t)$ at $t=t_k$. Finally, two examples are given to show the feasibility of our results.