## Differential and Integral Equations

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

Hal L. Smith

#### 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.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 8 (1995), 2125-2144.

Dates
First available in Project Euclid: 20 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1369056143

Mathematical Reviews number (MathSciNet)
MR1348968

Zentralblatt MATH identifier
0835.34071

#### Citation

Smith, Hal L. On the existence and stability of bounded almost periodic and periodic solutions of a singularly perturbed nonautonomous system. Differential Integral Equations 8 (1995), no. 8, 2125--2144. https://projecteuclid.org/euclid.die/1369056143