The Number Of Limit Cycles For A Class Of Planar Piecewise Linear Dynamical Systems

In this paper,the number of limit cycles for a class of planar piecewise linear differential dynamical systems with two regions is studied when each subsystem of the two regions has a saddle point.The main contents are arranged as follows: Chapter 1 to Chapter 4 mainly discusses the number of limit cycles when the discontinuous boundary(or switching boundary)is a straight line;In chapter 5,based on the previous four chapters,the influence of switching boundary from a straight line to two rays starting from the same point on the existence and number of limit cycles is further studied from the perspective of perturbation.The switching boundary is called the nonregular switching boundary,and the number of limit cycles under the nonregular switching boundary is discussed.The sixth chapter summarizes the whole paper,and gives the shortcomings of this paper and the future research direction.For the case where the switching boundary is a straight line,we analyze the left and right branches of the system(called the left and right half systems or the left and right subsystems)respectively,by constructing appropriate Poincaré mapping,and then discuss the inverse of the right Poincaré mapping and the left Poincaré.The number of intersection points of the mapping gives the complete result of the existence and number of limit cycles.This paper proves that there are at most two limit cycles in this type of system,and gives the detailed parameter ranges of the existence of limit cycles in each case.In the case of nonregular switching boundaries,the two section maps induced by the left and right subsystems at this time are composed of two different parts,each part is defined by an implicit parameter expression,and one of them may have an inflection point due to disturbance.At the end of Chapter 5,we use two specific numerical examples to illustrate that the emergence of inflection points plays an important role in generating more limit cycles than straight-line switching boundaries.Based on the results of this paper and the existing research results,it is obtained that the piecewise linear dynamical system with two real saddle-point subsystems may have at most two limit cycles when the switching boundary is a straight line.When the boundary of the first quadrant is used as the switching boundary,the maximum value of the limit cycle of the plane piecewise linear dynamical system with two real saddle-point subsystems exceeds 3.