# Asymptotic behavior of periodic solutions in one-parameter families of Li\'{e}nard equations

## TL;DR

This paper investigates how periodic solutions of Lie9nard equations behave asymptotically as the parameter varies, using relaxation oscillation and averaging theories for large and small parameter values.

## Contribution

It establishes a connection between relaxation oscillation and averaging theories to analyze the asymptotic behavior of solutions across the entire parameter range.

## Key findings

- Periodic solutions exhibit specific asymptotic behaviors for small and large e9ta.
- The paper links relaxation oscillation and averaging theories.
- Results apply to a broad class of Lie9nard equations.

## Abstract

In this paper, we consider one--parameter ($\lambda>0$) families of Li\'enard differential equations. We are concerned with the study on the asymptotic behavior of periodic solutions for small and large values of $\lambda>0$. To prove our main result we use the relaxation oscillation theory and a topological version of the averaging theory. More specifically, the first one is appropriate for studying the periodic solutions for large values of $\lambda$ and the second one for small values of $\lambda$. In particular, our hypotheses allow us to establish a link between these two theories.