Existence of a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle

The existence of a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle is proved, which is based on a novel decomposition of the dynamical system from the perspective of mechanics and some definitions and known theorems. By considering the smooth simple closed curve in the complex plane corresponding to the limit cycle of a smooth planar dynamical system and the definition of Morse decomposition, we first prove a theorem in which the limit cycle is diffeomorphic to the unit circle for any smooth planar dynamical system with one limit cycle, and we deduce another theorem, that any two smooth planar systems with one limit cycle are diffeomorphic (or smoothly equivalent). Next, through the definition of the potential function, the explicit construction of a smooth Lyapunov function for a smooth planar dynamical system with one circular limit cycle is given. Then, according to these results, we obtain the following theorem: there always exists a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle. Additionally, two examples are given. Finally, with respect to the coexistence of the limit cycle and Lyapunov function, we discuss two criteria related to system dissipation (divergence and dissipative power) in an example, find that they are not consistent, and explain the meaning of dissipation in infinitely repeated motion in the limit cycle. This result may provide a deeper understanding of the existence of a Lyapunov function for systems with limit cycles.