Asymptotic stability and periodic solutions of vector Li\'enard equations

TL;DR

This paper establishes the asymptotic stability of equilibrium solutions and the existence of periodic solutions in vector Li'enard equations using LaSalle invariance and intersection hypotheses.

Contribution

It introduces new stability and existence results for vector Li'enard equations under specific manifold intersection conditions.

Findings

Proves asymptotic stability of equilibrium solutions.

Deduces existence of periodic solutions under periodic perturbations.

Utilizes LaSalle invariance principle for stability analysis.

Abstract

We prove the asymptotic stability of the equilibrium solution of a class of vector Li\'enard equations by means of LaSalle invariance principle. The key hypothesis consists in assuming that the intersections of the manifolds in $\{\dot V = 0\}$ be isolated. We deduce an existence theorem for periodic solutions of periodically perturbed vector Li\'enard equations.