2019/03/11 by Xiaoliang Gan, Haoyu Wang, Gan, Xiao-Liang +6
Engineering · Mathematics · Physics and Astronomy · #34C-99(Secondary) #37B-25(Primary) #Advanced Differential Equations and Dynamical Systems #Advanced Thermodynamics and Statistical Mechanics #Control and Stability of Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #G.1.7 #I.6.1 #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1903.04690
openalex publication_date 2019/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A rigorous proof of a theorem on the coexistence of smooth Lyapunov function\nand smooth planar dynamical system with one arbitrary limit cycle is given,\ncombining with a novel decomposition of the dynamical system from the\nperspective of mechanics. We base on this dynamic structure incorporating\nseveral efforts of this dynamic structure on fixed points, limit cycles and\nchaos, as well as on relevant known results, such as Schoenflies theorem,\nRiemann mapping theorem, boundary correspondence theorem and differential\ngeometry theory, to prove this coexistence. We divide our procedure into three\nsteps. We first introduce a new definition of Lyapunov function for these three\ntypes of attractors. Next, we prove a lemma that arbitrary simple closed curve\nin plane is diffeomorphic to the unit circle. Then, the strict construction of\nsmooth Lyapunov function of the system with circle as limit cycle is given by\nthe definition of a potential function. And then, a theorem is hence obtained:\nThe smooth Lyapunov function always exists for the smooth planar dynamical\nsystem with one arbitrary limit cycle. Finally, by discussing the two criteria\nfor system dissipation(divergence and dissipation power), we find they are not\nequal, and explain the meaning of dissipation in an infinitely repeated motion\nof limit cycle.\n