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

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