A Class of Isochronous and Non-Isochronous Nonlinear Oscillators

TL;DR

This paper introduces a method to generate nonlinear oscillators with either amplitude-independent or dependent frequencies, providing a unified approach to derive their integrals and solutions, applicable to coupled systems.

Contribution

It presents a novel method for constructing nonlinear oscillators with specific frequency characteristics and derives their integrals and solutions in a unified framework.

Findings

Method applicable to 2N coupled nonlinear ODEs

Derivation of integrals and harmonic solutions

Illustrative examples demonstrating the approach

Abstract

In this work, we present a method of generating a class of nonlinear ordinary differential equations (ODEs), representing the dynamics of appropriate nonlinear oscillators, that have the characteristics of either amplitude independent frequency of oscillations or amplitude dependent frequency of oscillations from the integrals of the simple harmonic oscillator equation. To achieve this, we consider the case where the integrals are in the same form both for the linear and the nonlinear oscillators in either of the cases. We also discuss the method of deriving the associated integrals and the general solution in harmonic form for both the types. We demonstrate the applicability of this method up to 2N coupled first order nonlinear ODEs in both the cases. Further, we illustrate the theory with an example in each case.