Stable manifolds for impulsive delay equations and parameter dependence

Abstract

In this article, we establish the existence of Lipschitz stable invariant manifolds for the semiflows generated by the delay differential equation $x'= L(t)x_t + f(t,x_t,\lambda)$ with impulses at times $\{\tau_i\}_{i=1}^\infty $, assuming that the perturbation $f(t,x_t,\lambda)$ as well as the impulses are small and the corresponding linear delay differential equation admits a nonuniform exponential dichotomy. We also show that the obtained manifolds are Lipschitz in the parameter $\lambda$.

Keywords

• Delay impulsive equation
• exponential dichotomy
• stable invariant manifold