On the existence and stability of bounded almost periodic and periodic solutions of a singularly perturbed nonautonomous system

Abstract

The existence of solutions which are bounded on $ \mathbb{R}$, almost periodic or periodic is considered for a nonautonomous, singularly perturbed system of ordinary differential equations. In addition, the stability properties of these solutions are characterized by the construction of manifolds of initial data, the solutions for which approach the given solutions as $t\to+\infty$ ($t \to-\infty$) at an exponential rate, $\alpha$, independent of the small parameter. The key hypotheses are that certain linear systems have exponential dichotomies on $\mathbb{R}$. Applications are made to traveling wave solutions of reaction diffusion systems which are ``forced" by a traveling wave input.