On the linearization of isochronous centre of a modified Emden equation with linear external forcing

TL;DR

This paper develops a method to linearize an isochronous center in a modified Emden equation with external forcing, using Darboux integrals and transverse systems, revealing underlying mathematical structures.

Contribution

It introduces a novel approach to linearize isochronous centers in nonlinear systems via Darboux method and transverse commuting systems, providing explicit transformations.

Findings

Successfully linearized the modified Emden system

Constructed inverse integrating factor and first integral

Identified mathematical structures related to the system

Abstract

In this work, we carry out a detailed study on the linearization of isochronous centre of a modified Emden equation with linear external forcing. We construct inverse integrating factor and time independent first integral for this system through Darboux method. To linearize the isochronous centre we explore a transverse commuting dynamical system and its first integral. With the help of first integrals of the original dynamical system and its transverse commuting system we derive the linearizing transformation and reduce the nonlinear system into linear isochronous one. We also point out certain mathematical structures associated with this dynamical system.