Symmetric Liapunov center theorem

TL;DR

This paper generalizes the Liapunov center theorem for systems with symmetric potentials using an equivariant Conley index, establishing the existence of periodic solutions near non-degenerate stationary orbits.

Contribution

It introduces a novel approach employing equivariant Conley index to extend Liapunov center theorem to symmetric potential systems.

Findings

Existence of periodic orbits near stationary solutions with symmetric potentials.

Application of equivariant Conley index in infinite-dimensional settings.

Generalization applicable to systems with compact Lie group symmetries.

Abstract

In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider the system $\ddot{q}= -\nabla U(q),$ where $U(q)$ is a $\Gamma$-symmetric potential, where $\Gamma$ is a compact Lie group acting linearly on $\mathbb{R}^n$. If the system possess a non-degenerate orbit of stationary solutions $\Gamma(q_0)$ with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian $\nabla^2 U(q_0)$, then in any neighbourhood of orbit $\Gamma(q_0)$ there is a periodic orbit of solutions of the system.