Periodic Solutions Of An Indefinite Singular Equation

Singular equations with indefinite weights are an important class of differential equation models.In this dissertation,the dynamic properties of periodic solutions of singular equation with indefinite weight derived from Kepler problem on a spherical surface are studied.For indefinite with two singular points and a piecewise weight function,a new method is established to find exact periodic solutions.The main structure of this dissertation is as follows:In Chapter 1,the relevant research background and development status of singular equation with indefinite weight are introduced,and the main content and research characteristics of this dissertation are briefly described.In Chapter 2,the existence of periodic solutions for singular equations with indefinite weights is discussed.First,a series of transformations are constructed to transform the equation into an equivalent two-dimensional plane system.Next,it is proved that the mapping of planar system is definable in a certain region and the solution of the system exists.Finally,the necessary and sufficient conditions for the existence of periodic solutions are obtained.In Chapter 3,the stability analysis of periodic solution is given.It is testified that the stability of periodic solution is equivalent to the stability of fixed point of Poincarémapping,and then the stability of fixed point of Poincaré mapping is analyzed.In Chapter 4,numerical analysis is carried out for the singular equation with indefinite weight.For a specific example,the two periodic solutions of the system are solved by numerical calculation and its stability is analyzed.In Chapter 5,the results of this paper is summarized,and further prospects for the study of singular equations with indefinite weights are put forward.Compared with the results obtained by predecessors,the innovation of this dissertation lies in the innovation of methods and results.Using a new method,the exact periodic solution of the equation is found,and the stability of this periodic solution is discussed and analyzed.