Resonances and bifurcations in systems with elliptical equipotentials

TL;DR

This paper analyzes the orbit structures and bifurcations in 2D elliptical potentials using Lie transform normalization, revealing how resonances influence orbit stability and the emergence of specific orbit families.

Contribution

It provides a comprehensive analysis of resonances and bifurcations in elliptical equipotential systems, including stability conditions and the effects of deformations.

Findings

1:1 resonance leads to loop formation and mode destabilization

1:2 resonance results in stable bananas and unstable anti-bananas

Higher-order resonances occur at elevated energies and are well-characterized

Abstract

We present a general analysis of the orbit structure of 2D potentials with self-similar elliptical equipotentials by applying the method of Lie transform normalization. We study the most relevant resonances and related bifurcations. We find that the 1:1 resonance is associated only to the appearance of the loops and leads to the destabilization of either one or the other normal modes, depending on the ellipticity of equipotentials. Inclined orbits are never present and may appear only when the equipotentials are heavily deformed. The 1:2 resonance determines the appearance of bananas and anti-banana orbits: the first family is stable and always appears at a lower energy than the second, which is unstable. The bifurcation sequence also produces the variations in the stability character of the major axis orbit and is modified only by very large deformations of the equipotentials. Higher-order resonances appear at intermediate or higher energies and can be described with good accuracy.