Analysis And Application Of Singularity

Singularity analysis is an important content of the theoretical qualitative analysis of differential equation, and is very wide in application of various differential models. This paper mainly use the methods of qualitative theory to investigate the solutions or bifurcation of some models.The full paper includes five chapters. The first chapter mainly introduces the significance of the singularity analysis and status of foreign researches for the subject in differential equations.In the second chapter, we introduce the preliminary knowledge, which are the basic definitions, theorems and formulas used in third, four and five chapter.In the third chapter, we study the traveling wave solutions for a class of biological invasion model. Firstly by calculating singular point quantities, we determine the integrability of the system, and constructs the first integral of Hamilton type and applying phase diagram analysis to for solve the equation, then using the bifurcating methods of qualitative theory of dynamical system, we get some bounded traveling wave solutions of explicit expressions.In the fourth chapter, we discuss existence of bounded traveling wave for a class of (2+1) dimensional shallow water wave system, first through the analysis of the singularity of the dynamical system, phase portraits corresponding system were discussed, the phase diagram in the parameter are obtained under the condition of different classification. Finally, the bounded traveling wave solutions, including solitary wave solutions and periodic wave solutions are proved to exist.In the fifth chapter, we study the bifurcation of limit cycles of Chen system, applying a new algorithm for singular point quantity calculation of equilibrium points, for the system of two symmetric singular point respectively on the center manifold, the first2fine focal quantities are obtained, so as to further the conclusions are proved that at least4limit cycles can bifurcate for Chen system.