Existence of nonlinear normal modes for coupled nonlinear oscillators

TL;DR

This paper proves the existence of nonlinear normal modes in coupled nonlinear oscillators, demonstrating exact solutions for in-phase and out-of-phase periodic vibrations using the comparison principle.

Contribution

It establishes the existence of nonlinear normal modes for general coupled nonlinear oscillators, providing a rigorous mathematical proof.

Findings

Existence of exact in-phase and out-of-phase solutions

Application of the comparison principle to nonlinear oscillators

Identification of spatially localized time-periodic solutions

Abstract

We prove the existence of nonlinear normal modes for general systems of two coupled nonlinear oscillators. Facilitating the comparison principle for ordinary differential equations it is shown that there exist exact solutions representing a vibration in unison of the system. The associated spatially localised time-periodic solutions feature out-of-phase and in-phase motion of the oscillators.