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

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.
Comment: 21 pages,2 figures