Limit Cycle Theory And Application Of Several Planar Filippov Systems

In this article,the limit cycle problem of planar sliced linear differential systems with node-focus structure,node-node structure and focus-focus structure is analyzed by using the mathematical theory and method of discontinuous differential equation and differential inequality at the right end and the Poincaré mapping.A class of Morris-Lecar models,including the number and stability of limit cycles for planar sharded linear systems with node-focus structures,are discussed.The bounded properties of the solution of the Morris-Lecar model,the properties of the equilibrium point,the dynamic properties of the sliding mode on the switching line,and the limit cycles of the node-node type and focus-focus type of the system are also discussed.The full text consists of the following four parts.In the first chapter,the research background,development status and significance of limit cycles and neuron models for planar partitioned linear systems are introduced.We then introduce the main research contents and methods,and briefly state some basic theoretical knowledge used.In the second chapter,some basic mathematical concepts and theoretical knowledge to be used in this paper are given,mainly the qualitative and stability theory of Filippov system.In chapter 3,the limit cycle problem of planar sharded linear systems with nodefocus structure is introduced,including the number and stability of limit cycles.First,the canonical form of the node-focus type system with only 5 parameters is obtained by using nonsmooth homomorphism mapping.Secondly,on the basis of Poincare mapping,a successor function with a good structure is defined and its properties are given.Then,the main results of limit cycles for planar sliced linear systems with node-focus structures are given by using the properties of subsequent functions.The parameter regions with at least one limit cycle or two limit cycles,and the parameter regions with unique limit cycles or exactly two limit cycles are discussed,and the stability of limit cycles is analyzed.The correctness of the results obtained in this chapter is further verified by several numerical examples.In chapter 4,we mainly study the limit cycle problem of planar sharded linear systems with focus-focus structures and node-node structures.Using the obtained results,we further discuss the global dynamics of a class of Morris-Lecar models.The model is a plane-segmented linear system,which is firstly transformed into a canonical type with only five parameters.The dynamics of the subsystem and the sliding mode are obtained by using Lyapunov method and Filippov convex method,respectively.Similarly,we use the method in Chapter 3 to transform the limit cycle problem into a problem of finding the zero of a function.Using the properties of the successive functions,we give the results of the number of limit cycles and their stability of the node-node type and the focus-focus type.At the end of the article,our research work is summarized and the future research direction is prospected.