# Nonlocal transformations of the Generalized Li\'enard type equations and dissipative Ermakov-Milne-Pinney systems

## TL;DR

This paper develops a method using nonlocal transformations to linearize generalized Lie4nard equations, mapping them to dissipative Ermakov-Milne-Pinney systems, and derives new invariants and conditions related to isochronicity.

## Contribution

It introduces a generalized nonlocal transformation approach to linearize and analyze Lie4nard equations and their relation to Ermakov-Milne-Pinney systems, extending invariants and conditions.

## Key findings

- Mapped generalized Lie4nard equations to dissipative Ermakov-Milne-Pinney equations.
- Derived new first integrals for these equations.
- Established a relation between f(x) and g(x) similar to isochronicity conditions.

## Abstract

We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Li\'enard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov-Lewis invariant for a time dependent oscillator to (coupled) Li\'enard and Li\'enard type equations. We also study the linearization problem for the coupled Li\'enard equation using nonlocal transformations and derive coupled dissipative Ermakov-Milne-Pinney equation. As an offshoot of this nonlocal transformation method when the standard Li\'enard equation, x + f(x)x_ + g(x) = 0, is mapped to that of the linear harmonic oscillator equation we obtain a relation between the functions f(x) and g(x) which is exactly similar to the condition derived in the context of isochronicity of the Li\'enard equation.