# Reduction of Kinetic Equations to Li\'enard-Levinson-Smith Form: Counting Limit Cycles

## TL;DR

This paper introduces a unified method to convert various two-variable systems into Liénard-Levinson-Smith form, deriving conditions for limit cycles and using the Krylov-Bogoliubov method to determine their maximum number, demonstrated on multiple models.

## Contribution

It provides a novel unified scheme for transforming systems into LLS form and analyzing limit cycles, including conditions and maximum cycle count, applicable to polynomial damping and restoring forces.

## Key findings

- Derived conditions for limit cycles in LLS systems.
- Implemented Krylov-Bogoliubov method for maximum limit cycles.
- Validated scheme on multiple model systems.

## Abstract

We have presented an unified scheme to express a class of system of equations in two variables into a Li\'enard-Levinson-Smith (LLS) oscillator form. We have derived the condition for limit cycle with special reference to Rayleigh and Li\'enard systems for arbitrary polynomial functions of damping and restoring force. Krylov-Boguliubov (K-B) method is implemented to determine the maximum number of limit cycles admissible for a LLS oscillator atleast in the weak damping limit. Scheme is illustrated by a number of model systems with single cycle as well as the multiple cycle cases.