# Does the smooth planar dynamical system with one arbitrary limit cycle always exists smooth Lyapunov function?

## TL;DR

This paper proves that a smooth Lyapunov function always exists for any smooth planar dynamical system with a single arbitrary limit cycle, using a novel decomposition and classical mathematical theorems.

## Contribution

It provides a rigorous proof of the existence of smooth Lyapunov functions for systems with arbitrary limit cycles, combining mechanics-based decomposition and classical theorems.

## Key findings

- Existence of smooth Lyapunov function for any smooth planar system with one limit cycle.
- New decomposition method of dynamical systems from a mechanics perspective.
- Clarification of dissipation criteria in systems with limit cycles.

## Abstract

A rigorous proof of a theorem on the coexistence of smooth Lyapunov function and smooth planar dynamical system with one arbitrary limit cycle is given, combining with a novel decomposition of the dynamical system from the perspective of mechanics. We base on this dynamic structure incorporating several efforts of this dynamic structure on fixed points, limit cycles and chaos, as well as on relevant known results, such as Schoenflies theorem, Riemann mapping theorem, boundary correspondence theorem and differential geometry theory, to prove this coexistence. We divide our procedure into three steps. We first introduce a new definition of Lyapunov function for these three types of attractors. Next, we prove a lemma that arbitrary simple closed curve in plane is diffeomorphic to the unit circle. Then, the strict construction of smooth Lyapunov function of the system with circle as limit cycle is given by the definition of a potential function. And then, a theorem is hence obtained: The smooth Lyapunov function always exists for the smooth planar dynamical system with one arbitrary limit cycle. Finally, by discussing the two criteria for system dissipation(divergence and dissipation power), we find they are not equal, and explain the meaning of dissipation in an infinitely repeated motion of limit cycle.