The Existence Of Singular Cycles In Some Classes Of 3-dimensional Piecewise Affine Systems

The existence of homoclinic orbits or heteroclinic cycles plays a crucial role in the research of chaos since a lot of chaotic phenomena can be associated with them.For example,the famous Shil'nikov theorem and its extensions show that,under certain conditions,the existence of homoclinic orbits or heteroclinic cycles means the existence of horseshoes.However,it is often difficult to prove the existence of singular cycles in concrete systems,thus the corresponding existences were just given by assumption in these theorems.For piecewise affine system,its stable manifold,unstable manifold and analytic solution are all easy to be determined.Therefore,it is feasible to construct singular cycles by analytic method in some piecewise affine systems.In this paper,we concentrate on the existence of singular cycles in several classes of 3-dimensional piecewise affine systems,which have one or two switching planes.The work contains the following five chapters:In Chapter 1,the background and research status of singular cycles are briefly given.Meanwhile,we introduce the main result in this paper.In Chapter 2,the existence of homoclinic orbits in some classes of 3-dimensional piecewise affine systems with only one switching plane is discussed.Firstly,for a class of 3-dimensional piecewise affine systems without saddle-focus type equilibrium point,we give a very simple sufficient and necessary condition for the existence of homoclinic orbit.However,it will become quite difficult to build valid analytic conditions for the existence of homoclinic orbit when system have two saddle-focal type equilibria.To do this,we obtain some analytic conditions through some additional constraints.Moreover,the corresponding systems will display chaos under suitable conditions.In Chapter 3,we investigate the existence of homoclinic orbits in three classes of 3-dimensional three-zone piecewise affine systems with two switching planes,which have only one saddle-focus type equilibrium point.In particular,some sufficient conditions for the existence of two homoclinic orbits is obtained by an analytic method.Additionally,we construct some examples to illustrate the effectiveness of the method.In Chapter 4,we discuss the existence of heteroclinic cycles in some classes of 3-dimensional three-zone piecewise affine systems with two switching planes.Some sufficient conditions for the existence of a single or two heteroclinic cycles in three different cases are obtained.Likewise,we give some examples to illustrate our theoretical results.In Chapter 5,some concluding remarks and future research plans are given.