Dynamic Behavior Analyses Of A New Class Of Three-dimensional System And Its Corresponding Piecewise System And Exponential System

The meteorologist Lorenz found the first three-dimensional chaoticsystem-Lorenz system in1963. After that the chaotic dynamics have beenstudied deeply for years and the influence of nonlinear chaotic dynamic to thereal world is more far-reaching. The three-dimensional chaotic systemgroup-Lorenz system family is throughout the development process of chaoticdynamics.The original chaotic system in this paper belongs to Lorenz systemfamily. The corresponding piecewise system and expontial system of theoriginal system show the complex dynamic behaviors with the same initialvalues and minimal perturbation in the initial values.A new three-dimensional quadratic system is proposed in this paper,which is named as the primitive system. The system has the similar form ofLorenz system, Chen system and Lüsystem, but it is not topological equivalenceto the systems above in essence. By using the methods of Lyapunov exponents,bifurcation analysis theory and numerical simulations, its nonlinear characteristics have been revealed. At the same time, the composite structure ofthe new system is analyzed. Moreover, we add the nonlinear controller to thesystem and study the controlled system by the theory of Lyapunovcoefficient.On the basis of the original system, the new three-dimensionalpiecewise system is formed by changing the nonlinear terms in the originalsystem. The piecewise system has the unique attractor structure. Through thecombination of theoretical analysis and numerical simulation, we get the uniquedynamics of the corresponding piecewise system and discuss the possibility ofcircuit realization with the system. For the more, by changing the nonlinear termto the exponential term, the new exponential system has been founded. Theexponential system is similar to the original system with its unique feature.Similarly, by using the methods of Lyapunov exponents, bifurcation analysistheory and numerical simulation its nonlinear characteristics have been revealed.At the same time, the composite structure of the new system is analyzed.Moreover, we add the nonlinear controller to the system and study the controlledsystem by the theory of Lyapunov coefficient. For the special parameters in theoriginal system and the disturbance directly of the original system, the existenceof homoclinic orbits and periodic orbits are obtained and the parametricconditions can be given to lead the system from chaos to the low periodic track.The numerical simulations are also explained clearly to the conclusions.This paper is divided into seven chapters and the specific contents are asfollows.In the first chapter, the significance of the topic and the domestic andinternational development situation of this paper are briefly reviewed. In the endof this chapter, the main content of this paper is introduced.The definitions of chaos and bifurcation are introduced in the secondchapter. The main methods used in this paper to study the three-dimensionalsystems are introduced, such as the bifurcation diagram, phase diagram,Poincare section, time series, amplitude spectrum, Lyapunov index, correlation dimension, Lyapunov coefficients and generalized Melnikov method.In the third chapter, the new three-dimensional quadratic system ispointed out and studied. As the derivative system of the original system, thesystem has the similar form of Lorenz system, Chen system and Lüsystem, butit is not topological equivalence to the systems above essencely. Through usingthe methods of Lyapunov exponents, bifurcation analysis theory and numericalsimulations, its nonlinear characteristics have been revealed. The compositestructure of the new system is discussed. Moreover, adding the nonlinearcontroller to the system, we study the controlled system by the theory ofLyapunov coefficient and make some conclusions. We discuss the possibility ofcircuit realization with the system.In the fourth chapter, the new three-dimensional piecewise systemcorresponding to the original system is builded up by changing the nonlinearterms in the original system. The piecewise system has the unique attractorstructure.Through the combination of theoretical analysis and numericalsimulation, we get the unique dynamics of the corresponding piecewise systemand discuss the possibility of circuit realization with the system.In the fifth chapter, through changing the nonlinear term to theexponential term, the new exponential system has been founded and discussed.The exponential system is similar to the original system with its unique feature.Similarly, by using the methods of Lyapunov exponents, bifurcation analysistheory and numerical simulation its nonlinear characteristics have been revealed.At the same time, the composite structure of the new system is analyzed.Moreover, we add the nonlinear controller to the system and study the controlledsystem by the theory of Lyapunov coefficient.The special parameters in the original system and the disturbance directlyof the original system are considered in the sixth chapter. The systems havesimple zeros which are satisfied the parameter conditions, and for the more, thecorresponding numerical simulation diagrams are given. The seventh chapter summarizes the main work of this paper, at the sametime, the existing problems and the development trend of future are pointed out.