Abstract
This paper is devoted to the study of stable and unstable behaviors with growth rates $e^{c\rho(t)}$ for impulsive differential equations, where $\rho(t)$ is some increasing continuous function. By the techniques of impulsive analysis, we obtain the existence of stable invariant manifolds for the impulsive perturbed equation provided that the linear equation admits a ρ-nonuniform exponential dichotomy and f, g are sufficiently small Lipschitz perturbation. We also consider the case of exponential contraction and show that the asymptotic stability persists under sufficiently small nonlinear perturbations. In addition, we study how the manifolds vary with the perturbations.
Highlights
1 Introduction The theory of impulsive differential equations is attracting much attention in recent years. This is mostly because impulsive differential equations efficiently describe many phenomena arising in engineering, physics, and science as well
It is well known that the notion of exponential dichotomy, going back to Perron in [ ], plays an important role in invariant manifolds of differential equation
There are a lot of linear differential equations with
Summary
The theory of impulsive differential equations is attracting much attention in recent years. In [ ], Barreira and Valls studied the existence of stable invariant manifold for the linear equation under any sufficiently small nonlinear perturbation f. In [ ], the existence of invariant stable manifolds is established under sufficiently small perturbations of a linear equation and they showed that the invariant manifolds are of class C outside the impulsive points. General stable and unstable behaviors with growth rates of the form ecρ(t) for a function ρ(t) are exhibited by Barreira and Valls [ – ] This type of behavior is called a ρ-nonuniform exponential dichotomy. In [ ] the authors showed that for ρ in a large class of rate functions, any linear equation ( ) in a finite-dimensional space has a ρ-nonuniform exponential dichotomy.
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