| The research on the stability of differential dynamical systems is a concern in natural science and engineering technology.Among them,limit cycles occupies a very important position in the qualitative theory of differential equations.The research on it is both interesting and difficult.For example,the 16 th problem of 23 mathematical problems that mathematicians should work hard to solve in the new century proposed by Hilbert in 1900 involves the existence and distribution of limit cycles of planar polynomial systems.The research on qualitative analysis of systems has developed differently.Among them,the Lyapunov function is one of the most popular methods,but it depends on its existence.This doctoral dissertation will focus on the existence of Lyapunov function of limit cycle systems,and mainly do three aspects of work:First of all,as people continue to deepen the research and application of differential dynamical systems,many problems involve limit cycles,an important type of attractor,but surprisingly,the problem of the existence of a Lyapunov function of limit cycle systems has always been confused the researchers.In Chapter 3,the existence of a smooth Lyapunov function for any smooth planar dynamical system with exactly one limit cycle and no finite singularity outside the 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 exactly one limit cycle and no finite singularity outside the limit cycle,and further deduce another theorem that any two smooth planar systems with exactly one limit cycle and no finite singularity outside the limit cycle are diffeomorphic(or smoothly equivalent).Next,through the definition of the potential function,the explicit construction of the smooth Lyapunov function for a smooth planar dynamical system with exactly 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 exactly one limit cycle and no finite singularity outside the 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 the system’s 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.Next,in the discussion part of Chapter 3,we found that the two criteria(divergence and dissipative power)for determining the system’s dissipation are inconsistent,and found that the use of divergence to determine the system’s dissipation is problematic and contradictory.In Chapter 4,a criterion,dissipative power,beyond divergence for determining the dissipation of a system is presented,which is based on the knowledge of classical mechanics and a novel dynamic structure by Ao.Moreover,the relationship between the dissipative power and potential function(or called Lyapunov function)is derived,which reveals a very interesting,important,and apparently new feature in dynamical systems: to classify dynamics into dissipative or conservative according to the change of “energy function” or “Hamiltonian”,not according to the change of phase space volume.We start with two simple examples corresponding to two types of attractors in planar dynamical systems: fixed points and limit cycles.In determining the dissipation by divergence,these two systems have both the elusive contradictions pointed by researchers and new ones noticed by us.Then,we combine these two examples,analyze and compare these two criteria,further consider the planar linear systems with the coefficient matrices being the four types of Jordan’s normal form,and find that the dissipative power works when divergence exists contradiction.Finally,the obtained relationship between the dissipative power and the Lyapunov function provides a reasonable way to explain why some researchers think that the Lyapunov function does not coexist with the limit cycle.Those results may provide a deeper understanding of the dissipation of dynamical systems.Finally,we comment on a recent article [Rodriguez-Sanchez et al.,PLo S Comput.Biol.,16(4): e1007788(2020)].On the one hand,Rodriguez-Sanchez et al.used“Escher’s stairs” as a metaphor for cyclic attractors,and then claimed that there was no potential landscape in the limit cycle.On the other hand,they also introduced a decom position based on Helmholtz’s idea,which decomposed the vector field into conservative or gradient parts and non-gradient parts.Based on the artistic metaphor and the decomposition,Rodriguez-Sanchez et al.thought that the potential function of a system is only determined by the gradient part,and then point out that when the non-gradient term is large,the potential function cannot be found.On this point,they gave examples where potential landscapes cannot be solved by their decomposition,such as the linear system with equilibrium point as the center and the Lotka-Volterra model with a limit cycle.According to Ao’s novel decomposition of dynamics,we show that the evolution of a system is the result of the interaction between gradient and Hamilton dynamics,and reveal the misleading of Escher’s stair metaphor and Rodriguez-Sanchez’s incorrect decomposition of potential function only determined by gradient. |
