On The Problems Of Bifurcation Of Planar Piecewise Linear Dynamical Systems

The study of limit cycles of discontinuous dynamical systems is important both for theory and practical applications.Due to the existence of separation lines and the increase of number of parameters,the research of discontinuous dynamical systems is much more difficult than that of continuous dynamical systems.Even for discontinuous systems with a very simple form,the number of limit cycles is still difficult to determine.For example,for the planar piecewise linear dynamical systems whose separation line is a straight line,its upper bound of the number of limit cycles is still an open problem.We are interested in the reasons for the change of the limit cycles of discontinuous systems when the separation lines change,and also the way to change.Therefore,we study the limit cycles of the planar piecewise linear dynamical systems.We study three classes of planar piecewise linear dynamical systems.Their separation lines are either two non-collinear rays starting from the same point,or three rays starting from the same point and two of which are collinear,and every subsystem's singular point is focus or center or saddle.Poincar(?) map is used to find limit cycles of the three systems.We study the number of limit cycles,and find that the angle ? = 0or ? of piecewise region is the critical value.That is,by comparing with the planar piecewise linear dynamical systems separated by a straight line,we get that when the angle of piecewise region is not 0 or ?,or the number of the separation rays increase,the number of the limit cycles of planar piecewise linear dynamical systems increase under the same conditions.We remark that “the same conditions” refers to the types and the relative positions of singular points of the subsystems.However,why does the number of limit cycles increase? We obtain the answer of the question,which is the change of the complexity of the Poincar(?) map caused by the change of the the number of pieces or the angle of the separation rays.We study the three planar piecewise linear dynamical systems in three sections.In the first section,we study a class of Y-shape planar piecewise linear dynamical systems with focus-focus-focus type.Their separation lines are three rays starting from the same point and two of which are collinear,and every subsystem's singular point is focus.We first obtain the canonical form of this special Y-shape systems,which reduces the number of parameters in original systems from 16 to 10.Next,we demonstrate that the graph of Poincar(?) map of subsystem in piecewise region with angle ? ?= ? has at most one inflection point and can has one.Lastly,we prove that for some parameters there are 4 limit cycles for this special systems.Specifically,we get that when the singular point of subsystem in the piecewise region with angle ? = ? is a boundary focus or a center,the cyclicity of this Y-shape systems is 3.However,we from [26] know that when the singular point of one subsystem is a boundary focus or a center,the cyclicity of planar piecewise linear dynamical systems with focus-focus type separated by a straight line is 1.Hence,when the separation line is changed from a straight line to Y-shape rays,the number of the limit cycles of planar piecewise linear dynamical systems is increased under the same conditions.For planar piecewise linear systems with two pieces separated by a straight line,the graph of Poincar(?) map of each subsystem is a ray or a curve which is global concavity or global convexity,see [26].It is clear that the number of limit cycles of planar piecewise linear systems increases as the convexity or concavity of the graphs of Poincar(?) maps changes,and the change of Poincar(?) maps is caused by the change of the number of pieces or the angle of the separation rays.In the second section,we study a class of linear lateral dynamical systems with focus-focus type.Their separation lines are two non-collinear rays starting from the same point,and both subsystems' singular points are focus.By studying Poincar(?) maps of subsystems,we get that for some parameters there can exist 5 limit cycles for this special systems.This is the lower bound of the maximum number of limit cycles of linear lateral systems so far.However,for the planar piecewise linear dynamical systems separated by a straight line,the lower bound of the maximum number of limit cycles is 3,see Table 1.1.We also obtain that if one of two subsystems' singular point is the origin and the systems have no sliding set,then the cyclicity of systems is 2,and the systems can have 0,1 and 2 limit cycles for some parameters.We know from [26] that there is no limit cycle in the planar piecewise linear dynamical systems with focus-focus type separated by a straight line under the same conditions.These results clarify that if the separation line changes from a straight line to two non-collinear rays starting from the same point under the same conditions,then the number of limit cycles of planar piecewise linear dynamical systems with focus-focus type is increased.Obviously,the reason for the increase in the number of limit cycles is the change of the complexity of the Poincar(?) map which is caused by the change of the angle of separation rays.In the third part,we study a class of linear lateral dynamical systems with saddlecenter type.Their separation lines are two non-collinear rays starting from the same point,and one subsystem' singular point is saddle and the other is center.We demonstrate that the graph of Poincar(?) map of subsystem with saddle has at most one inflection point and can has one.Based on this,we get that when the singular points of the two subsystems are standard saddle and standard center,respectively,the cyclicity of the systems is 1.We also obtain that if the center of the subsystem is at origin,then the systems can have at most 3 limit cycles and there can exist 2 limit cycles for some parameters.By disturbing the singular point of subsystem with center,we can get that there can exist 3 limit cycles for the systems.However,J.Llibre and X.Zhang proved in [24] that the cyclicity of the planar piecewise linear dynamical systems with saddle-center type separated by s straight line is 1.Obviously,if the separation line changes from a straight line to two non-collinear rays starting from the same point under the same conditions,then the number of limit cycles of planar piecewise linear dynamical systems with saddle-center type is increased.S.Huan and X.Yang proved in [32] that the graph of Poincar(?) map of each subsystem of planar piecewise linear dynamical systems with saddle-saddle type separated by a straight line has no inflection point.Therefore,the reason for the increase in the number of limit cycles is the change of the complexity of the Poincar(?) map which is caused by the change of the angle of separation rays.