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A rigorous proof of a theorem on the coexistence of smooth Lyapunov functionand smooth planar dynamical system with one arbitrary limit cycle is given,combining with a novel decomposition of the dynamical system from theperspective of mechanics. We base on this dynamic structure incorporatingseveral efforts of this dynamic structure on fixed points, limit cycles andchaos, as well as on relevant known results, such as Schoenflies theorem,Riemann mapping theorem, boundary correspondence theorem and differentialgeometry theory, to prove this coexistence. We divide our procedure into threesteps. We first introduce a new definition of Lyapunov function for these threetypes of attractors. Next, we prove a lemma that arbitrary simple closed curvein plane is diffeomorphic to the unit circle. Then, the strict construction ofsmooth Lyapunov function of the system with circle as limit cycle is given bythe definition of a potential function. And then, a theorem is hence obtained:The smooth Lyapunov function always exists for the smooth planar dynamicalsystem with one arbitrary limit cycle. Finally, by discussing the two criteriafor system dissipation(divergence and dissipation power), we find they are notequal, and explain the meaning of dissipation in an infinitely repeated motionof limit cycle.

Does the smooth planar dynamical system with one arbitrary limit cycle always exists smooth Lyapunov function?