The Research On Complex Dynamics Of Singular Cycles And Chaos In Three & Four-dimensional Piecewise-smooth Systems

Chaos exists in nature widely,and its generation mechanism has become an important and key part of modern chaos research.It is well known that chaos cannot be generated by one-dimensional and two-dimensional smooth autonomous systems,and the global dynamic behavior of finite-dimensional linear smooth systems can be accurately predicted.However,it is very difficult to study the global dynamics of finite-dimensional nonlinear smooth systems,even though the chaotic behavior of three-dimensional quadratic polynomial systems,such as the classical Lorenz system,has not been fully understood so far,let alone three dimensional higher order system and a system with no less than quadratic in three and higher dimension.How to study complex chaos characteristics from a system with a simple form,the research on chaotic complexity of piecewise linear system consisting of linear equations,which provides a effective path and breakthrough in studying the generation mechanism of chaos,are of great significance to reveal the essence of chaos movement.On the one hand,the singular cycles including homoclinic and heteroclinc cycles in the Shilnikov chaotic theory offer a route to generate chaos and provide a significant idea of understanding the nature of chaos.On the other hand,the designed model based on piecewise linear system is easily operated and can present complex dynamic behavior.Thus,it has widespread application in the fields of circuit system,secure communication,mechanical engineering and so on.It is significant to study the singular cycles and chaotic dynamical behaviors of piecewise linear system in either theory or practical application.This paper investigates the complex dynamics of singular cycles and chaos in some three and four dimensional piecewise-smooth systems.By analyzing the stable manifold,the unstable manifold and the discontinuous boundary,this paper first establishes criterions for detecting homoclinic and heteroclinic cycles with overcoming the difficulties to which the transcendental functions give rise.Furthermore,based on the relevant theory of symbolic dynamical system and topological horseshoe,this paper strictly analyzes the dynamical behaviors of such system by constructing the Poincar?e map in a small neighbourhood of homoclinic and heteroclinic cycles,and establishes the existence conditions of chaos,i.e.,Shilnikov-type theorem.The Shilnikov-type theorem established in this paper is a supplement and expansion of Shilnikov theorem of smooth dynamical system in piecewise-smooth system.The detailed research work is as follows:Chapter 1 states the development process and research significance of chaos,and some classical chaotic definitions.The research status and progress of piecewise smooth systems are briefly introduced.And the concepts,definitions and propositions related to research work in this paper are reviewed.Chapter 2 investigates the dynamics of a class of three dimensional piecewise-smooth systems with two discontinuous boundaries.Three different types of heteroclinc cycles:(a)connecting two different saddle-focus equilibria;(b)connecting a saddle-focus equilibrium and a saddle equilibrium;(c)connecting two different saddle equilibria,are comprehensively discussed.The criteria for ensuring the existence of heteroclinic cycles with their corresponding types are respectively established,and rigorous mathematical proofs are provided.Furthermore,motivated by the ideal of Shilnikov theory,we show the existence of topological horseshoe by constructing and analyzing the Poincar?e map,and obtain the Shilnikov-type theorem to guarantee the existence of chaotic invariant set.Moreover,concrete examples are offered to illustrate that the simulation results obtained by numerical experiment are consistent with the theoretical results.Chapter 3 studies the dynamics of a class of three dimensional three-zone piecewisesmooth systems.The existence of homoclinic and heteroclinic cycles,coexistence of homoclinic and heteroclinic cycles,coexistence of two different homoclinic cycles are discussed,respectively.Sufficient conditions for the existence of singular cycles in different forms are established and strict mathematical proofs are given.Based on homoclinic cycle,heteroclinic cycle and their coexistence,the corresponding Shilnikov-type theorems are established,respectively.In particular,this paper gives a detailed analysis of the coexistence of singular rings and the chaotic behavior caused by coexistence.The correctness and validity of the theoretical results are verified by simulation and observation of numerical examples that satisfy the conditions of theorems.Chapter 4 researches the dynamics in a class of four dimensional piecewise-smooth system with one discontinuous boundary.The dynamic behaviors of such system along stable and unstable manifolds are analyzed in details.By changing the direction of the manifold,the system has a homoclinic cycle under certain conditions.Basing on the existence of homoclinc cycle,the Shilnikov-type theorem corresponding the four dimensional system is established,and a rigorous proof is shown.At the same time,the correctness of theoretical results and analysis process is illustrated by numerical simulation.