The symmetric 1:2 resonance

TL;DR

This paper applies Lie transform normal-form theory to analyze the 1:2 resonance in Hamiltonian systems, providing insights into bifurcations, stability, and phase-space structure, with applications to galactic potential models.

Contribution

It introduces a method to construct the 1:2 resonant normal form for natural Hamiltonian systems with symmetries, enabling detailed bifurcation and stability analysis.

Findings

Bifurcation points of main periodic orbits identified

Stability regions of normal modes characterized

Phase-space structure insights for galactic models

Abstract

This paper illustrates the application of Lie transform normal-form theory to the construction of the 1:2 resonant normal form corresponding to a wide class of natural Hamiltonian systems. We show how to compute the bifurcations of the main periodic orbits in a potential with double reflection symmetries. The stability analysis of the normal modes and of the periodic orbits in general position allows us to get overall informations on the phase-space structure of systems in which this resonance is dominating. As an example we apply these results to a class of models useful as galactic potentials.