Bifurcations of symmetric periodic orbits via Floer homology

TL;DR

This paper develops criteria using equivariant Floer homology to detect bifurcations of symmetric periodic orbits in reversible Hamiltonian systems, with applications to celestial mechanics.

Contribution

It introduces a Floer homology-based method for identifying bifurcations of symmetric periodic orbits, extending previous numerical approaches.

Findings

Bifurcation criteria established via Floer homology

Application to rotating Kepler problem shows bifurcations of torus orbits

Connection to Hénon's numerical work on Hill's lunar problem

Abstract

We give criteria for the existence of bifurcations of symmetric periodic orbits in reversible Hamiltonian systems in terms of local equivariant Lagrangian Rabinowitz Floer homology. As an example, we consider the family of the direct circular orbits in the rotating Kepler problem and observe bifurcations of torus-type orbits. Our setup is motivated by numerical work of H\'enon on Hill's lunar problem.