# A puzzle on the existence of Lyapunov functions for Limit cycle system

## TL;DR

This paper investigates the longstanding puzzle of whether Lyapunov functions exist for limit cycle systems, proposing a new definition and analyzing dissipation criteria through physical and mathematical perspectives.

## Contribution

It introduces a new equivalent definition of Lyapunov functions and analyzes dissipation criteria, offering insights into the existence of Lyapunov functions for limit cycle systems.

## Key findings

- Divergence is not zero while trajectories form infinite loops.
- Existing dissipation criteria are inconsistent in representing dissipation.
- Limit cycles can be viewed as isopotential lines for charged particles in electromagnetic fields.

## Abstract

Although the limit cycle have been studied for more than 100 years, the existence of its Lyapunov function is still poorly understood. By considering a common limit cycle system, a puzzle related to the existence of Lyapunov functions for Limit cycle system is proposed and studied in this paper: The divergence is not equal to zero, while the trajectory can be infinite loop on limit cycle. It will be discussed from three aspects. Firstly, the definition of Lyapunov function is concerned, and a new version of definition is given which is equivalent to the usual one and better adapts to the properties of Lyapunov function. Secondly, two criteria(divergence and dissipation power) are discussed from the perspective of dissipation, and it reaches that they are not consistent in representing the dissipation. Thirdly, by studying the motion of charged massless particle in electromagnetic field, it obtains that the limit cycle is an isopotential line on which the charged particles can move infinitely. Such discussions above may provide an understanding on the puzzle about the existence of Lyapunov functions for limit cycle system.