Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance

TL;DR

This paper reviews the bifurcation sequences of periodic orbits in symmetric Hamiltonian systems with nearly equal frequencies, using geometric and singularity theory methods, with applications mainly in astrodynamics.

Contribution

It provides a comprehensive geometric framework for analyzing bifurcations in symmetric Hamiltonian resonant systems, linking phase-space dynamics with catastrophe theory.

Findings

Characterization of bifurcation sequences in symmetric Hamiltonian systems

Development of an energy-momentum map for phase space analysis

Application of the framework to astrodynamics problems

Abstract

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under $Z_2 \times Z_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.