Dynamics of a Completely Integrable $N$-Coupled Li\'enard Type Nonlinear Oscillator

TL;DR

This paper introduces a class of N-coupled Lie9nard nonlinear oscillators that are completely integrable, possess explicit integrals, and can be transformed into harmonic oscillators, revealing diverse periodic and quasiperiodic solutions.

Contribution

It presents a new integrable N-coupled Lie9nard oscillator system with explicit integrals and a Hamiltonian structure, including special superintegrable cases.

Findings

System is completely integrable with explicit integrals.

Transformable to uncoupled harmonic oscillators via contact transformation.

Exhibits periodic, quasiperiodic, and decaying solutions depending on parameters.

Abstract

We present a system of $N$-coupled Li\'enard type nonlinear oscillators which is completely integrable and possesses explicit $N$ time-independent and $N$ time-dependent integrals. In a special case, it becomes maximally superintegrable and admits $(2N-1)$ time-independent integrals. The results are illustrated for the N=2 and arbitrary number cases. General explicit periodic (with frequency independent of amplitude) and quasiperiodic solutions as well as decaying type/frontlike solutions are presented, depending on the signs and magnitudes of the system parameters. Though the system is of a nonlinear damped type, our investigations show that it possesses a Hamiltonian structure and that under a contact transformation it is transformable to a system of uncoupled harmonic oscillators.