Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators

TL;DR

This paper analyzes the bifurcation of periodic solutions in a ring of coupled nonlinear oscillators, identifying symmetry properties and specific solution branches for different potentials using advanced mathematical tools.

Contribution

It provides a comprehensive analysis of bifurcations from relative equilibria in nonlinear oscillator rings, including explicit results for cubic and saturable potentials.

Findings

Bifurcation of solution branches from relative equilibria proven.

Complete results for cubic Schrödinger and saturable potentials.

Symmetry properties of bifurcating solutions analyzed.

Abstract

This paper gives an analysis of the periodic solutions of a ring of $n$ oscillators coupled to their neighbors. We prove the bifurcation of branches of such solutions from a relative equilibrium, and we study their symmetries. We give complete results for a cubic Schr\"odinger potential and for a saturable potential and for intervals of the amplitude of the equilibrium. The tools for the analysis are the orthogonal degree and representation of groups. The bifurcation of relative equilibria was given in a previous paper.