  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.
