A class of solvable coupled nonlinear oscillators with amplitude independent frequencies

TL;DR

This paper identifies and analyzes a class of coupled nonlinear oscillators that uniquely exhibit amplitude-independent frequencies, providing explicit solutions and demonstrating their periodic and quasiperiodic behaviors.

Contribution

It introduces a new class of coupled nonlinear oscillators with amplitude-independent frequencies and provides explicit solutions and reduction methods.

Findings

Existence of amplitude-independent frequencies in certain nonlinear oscillators

Explicit solutions for a subset of these oscillators

Reduction of systems to integrable scalar differential equations

Abstract

Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Li\'enard type oscillators exhibit this interesting property. We show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions. Another set of $N$-coupled nonlinear oscillators, possessing the amplitude independent nature of frequencies, is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically.