The Dynamics Of The Three-dimensional Piecewise Linear System With Double Saddle-focus

Piecewise smooth dynamic system is one kind typical system of nonlinear dynamic system,and the most common of the piecewise smooth dynamic system is piecewise lin-ear system.Many dynamical systems that occur naturally in the description of natural physical phenomena,such as switching circuits in power electronics.Piecewise linear sys-tems are also widely used in mechanical engineering,control biology,power electronics,aerospace and other fields.And,this kind of system have complicated nonlinear phe-nomenon,which can appear chaotic motion under certain conditions.Therefore,it is necessary to develop the theory of piecewise-line dynamical systems to study these prob-lems.And also,it has important theoretical and practical significance to explore internal movement mechanism and the dynamic behavior of the of piecewise linear system.In recent years,the research about the periodic motion,the stability,limit cycles,bifurca-tion and chaos of piecewise linear system has been a very important subject.Although,a lot of outstanding researchers have already made great progress under the unremitting efforts,there are still lots of problems of piecewise-line dynamical systems have not yet been solved.In this article,we mainly studied the three-dimensional piecewise linear system,and we analyzed the dynamic behavior of this system through both theoretical analysis and numerical simulation methods.The chaotic phenomenon in nonlinear system is very common,so it has the vital significance to research the chaos of piecewise linear system.This paper,we devoted ourselves to studying a class of piecewise-linear dynamic system with double saddle-focus and have obtained innovation results as follows.(1)We investigate the 3-dim piecewise linear dynamic system with double saddle-focus,strictly analyze the existence of heteroclinic loop and homoclinic orbits of this sys-tem,and give the corresponding theorem conditions.On this basis,we use the Shilnikov theorem strictly proved the existence condition of the chaotic attractor of the 3-dim piecewise-linear dynamic system.(2)we consider the conditions of periodic orbits of three-dimensional piecewise linear systems with double saddle-focus.We mainly deduce the theorem conditions of periodic orbits from theoretical analysis by constructing Poincare map.(3)We apply the methods obtained in the previous chapters to the modified Lorenz system,a specific three dimensional piecewise linear systems with double saddle-focus,and obtain the heteroclinic loop,homoclinic orbits,periodics and chaotic attractor under the condition of different parameters of the system.Finally,we calculate the Lyapunov exponents by using the computer,and the numerical simulation which consistent with the results of theoretical derivation that verify the validity of the theoretical derivation.