Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators

TL;DR

This paper investigates the global bifurcation phenomena of polygonal relative equilibria in systems of masses, vortices, and oscillators, utilizing symmetry analysis and degree theory to understand solution structures.

Contribution

It introduces a unified approach to analyze bifurcations in symmetric polygonal configurations across different physical models.

Findings

Identification of bifurcation points depending on a key parameter

Characterization of symmetry properties of bifurcated solutions

Application of degree theory to establish existence of solutions

Abstract

Given a regular polygonal arrangement of identical objects, turning around a central object (masses, vortices or dNLS oscillators), this paper studies the global bifurcation of relative equilibria in function of a natural parameter (central mass, central circulation or amplitude of the oscillation). The symmetries of the problem are used in order to find the irreducible representations, the linearization and, with the help of a degree theory, the symmetries of the bifurcated solutions.