The existence of Lyapunov function for the planar system with an arbitrary
limit cycle is proved. Firstly, the generalized definition of Lyapunov function
for fixed point and limit cycle are given, respectively. And they are logically
consistent with the definition in dynamical systems textbooks. Secondly,
combined with Schoenflies theorem, Riemann mapping theorem and boundary
correspondence theorem, that arbitrary simple closed curve in plane can be
mapped to the unit circle one by one is proved. Thirdly, according to the
definition of potential function in physics, the one-dimensional radial system
of polar coordinate system corresponding to two-dimensional dynamic system is
studied, and then the strictly analytic construction of Lyapunov function is
given for the system with an unit circle as a limit cycle. Finally, by
discussing two well-known criteria for system dissipation, that they are not
equivalent is demonstrated. Such discussion may provide an understanding on the
confusion on Lyapunov function in limit cycles still existing in recent
textbooks. What's more, some corresponding examples are provided above.