Limit Cycles And Chaos Of 3D Piecewise Affine Systems

Chaos is a phenomenon similar to random motion in a deterministic system.Chaos has a wide range of applications in secure communications,aerospace and other fields.However,how to prove the existence of chaos in a system is still a complex problem.Many scholars understand the chaos generation mechanism of complex systems by studying the chaos generation mechanism of simple systems.The piecewise affine has the characteristics of simple form and rich dynamics.Therefore,the study of piecewise affine systems can help understand the dynamics of complex systems.This paper studies the limit cycles and chaos of a class of 3D piecewise affine systems.Through the analysis of the system orbit,a sufficient condition for no sliding motion in the positive half orbit from the research area is obtained.At the same time,it is proved that the system has two non-sliding mode homoclinic orbits connecting saddle points.By selecting a suitable cross section,the system's Poincaré mapping is established.Based on the analysis of Poincare mapping,proves the existence of limit cycles,heteroclinic orbits and chaotic invariant sets after the homoclinic orbit breaks.The specific research content is as follows:The first chapter briefly describes the development of chaos and the significance of the study of piecewise affine systems.The definitions of Li-Yorke chaos,Devaney chaos and Wiggins chaos,as well as the symbolic dynamic system,Shilnikov theorem and Filippov system related to the research content are introduced.Chapter 2 proposes a type of 3D piecewise affine system.By analyzing the orbit of the system,it gives a sufficient condition for the system to start from a certain area without sliding motion in the positive half orbit.Combining the stable and unstable manifolds of the saddle,it is proved that the system has two non-sliding homoclinic orbits connecting the saddle.Furthermore,the correctness of the results is verified by numerical simulation.The third chapter mainly studies the existence of limit cycles and the types of limit cycles of the system after the homoclinic orbit breaks.By selecting a suitable cross section,the Poincare mapping of the system is obtained.Through the study of Poincare mapping,it can be known that the existence of the limit cycle of the system can be transformed into the existence of the periodic point of the one-dimensional mapping.Based on this fact,proved the existence of limit cycles and heteroclinic orbits under different bifurcation parameters.The correctness of the results is verified by numerical simulation.Chapter 4 mainly studies the existence of the chaotic invariant set of the system.Through the study of the Poincare mapping by the system,it is proved that the system has a Cantor invariant set.At the same time,a sufficient condition for the coexistence of the chaotic invariant set of the system and the two saddle type limit cycles is given.The correctness of the results is verified by numerical simulation.