Factorization technique and isochronous condition for coupled quadratic and mixed Li\'enard-type nonlinear systems

TL;DR

This paper presents a systematic method to factorize coupled quadratic and mixed Lie9nard-type nonlinear ODEs, including conditions for isochronous behavior and examples of physical systems.

Contribution

It introduces a new factorization procedure for coupled nonlinear equations and derives isochronous conditions for this class of systems.

Findings

Developed a systematic factorization method for coupled nonlinear ODEs.

Derived isochronous conditions for coupled quadratic Lie9nard equations.

Provided examples illustrating physical applications of the method.

Abstract

In this paper, we discuss a systematic and self consistent procedure to factorize a rather general class of coupled nonlinear ordinary differential equations (ODEs), namely coupled quadratic and mixed Li\'enard type equations, which include various physical and mathematical models. The procedure is broadly divided into two parts. In the first part, we consider a general factorized form for the equation under consideration in terms of some unknown functions and identify the determining equations for them. In the second part, we systematically solve the determining equations and identify the compatible factorizing form for this class of equations. In addition, we also discuss the problem of identification of isochronous dynamical systems belonging to the above class of equations. In particular, we deduce an isochronocity condition for the coupled quadratic Li\'enard equation. We also present specific examples of physical interest.