Existence of Lyapunov function for the planar system with one arbitrary limit cycle

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. Whatu0027s more, some corresponding examples are provided above.