Qualitative Theory Research On Some Special Lagrange Systems

In this paper,the singular points and their stability of some special Lagrange systems are studied based on equations theory of differential equations and method of differential gradient system.The trajectories in phase space and the dynamics behaviors of systems are studied by numerical simulation and plotting the Poincare surface of section,phase diagram and time-series diagram.Chapter one briefly introduces the history and current status of Lagrange system qualitative theory.Chapter two introduces several kinds of methods of judgment the stability of singular points.Chapter three mainly studies the stability of singular points of two degree of freedom autonomous Lagrange system by Lyapunov indirect method.Chapter four discusses the combined gradient system,differential equation of combined gradient system are given,and its properties and the application of judgement Lagrange system stability are studied.Chapter five analyzes the dynamic behavior of weak nonlinear coupled two-dimensional system.Firstly,a weak nonlinear coupled two-dimensional system is constructed by using coupling spring,Lagrange function and the differential equations of this system are established;then,the singular points for the system are obtained and their stability are judged by using the Lyapunov method;finally,the trajectories in phase space of the system are studied by numerical simulation and plotting the Poincare surface of section,phase diagram and time-series diagram.Chapter six studies the dynamic behavior of the weak nonlinear coupled two-dimensional anisotropic harmonic oscillator.The singular points and its stability of this system are studied;then,numerical simulations by Matlab and Poincare surface of section are used to study the trajectories of the system in phase space,and it is found that,with the increase of energy,the chaos appears finally.At last,we conclude the whole paper and show some expetations for future researches.