Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria

TL;DR

This paper proves the existence of periodic solutions near equilibrium orbits in symmetric Newtonian systems using equivariant bifurcation and Conley index techniques, generalizing the Lyapunov center theorem.

Contribution

It introduces an equivariant Conley index approach to extend the Lyapunov center theorem to systems with symmetric potentials and non-isolated critical orbits.

Findings

Existence of periodic solutions near equilibrium orbits in symmetric systems.

Application of equivariant bifurcation techniques to Newtonian systems.

Generalization of the Lyapunov center theorem for symmetric potentials.

Abstract

The aim of this paper is to prove the existence of periodic solutions to symmetric Newtonian systems in any neighborhood of an isolated orbit of equilibria. Applying equivariant bifurcation techniques we obtain a generalization of the classical Lyapunov center theorem to the case of symmetric potentials with orbits of non-isolated critical points. Our tool is an equivariant version of the Conley index. To compare the indices we compute cohomological dimensions of some orbit spaces.