On The Bifurcation Of A Class Of Nonlinear BVP Via Singularity Theory

In the bifurcation theory, studying the existence and the numbers of bifurcation solutions of a nonlinear differrntial equation with bifurcation parmeter is an important problem. It is also significant to study the effect of the nonlinear term on the existence and the numbers of bifurcation solutions of a nonlinear differrntial equation. This thesis mainly study the effect of bifurcation properties of nonlinear term on bifurcation properties of BVP (boundary value problem).The content of this paper will be mainly divided into four parts. In the first chapter, we briefly introduce the developmental course of singularity theory and mainly researches. Next, we introduce the application of singularity theory in bifurcation theory, then briefly introduce researches and innovations of this paper. In the second part of this paper, we briefly introduce some definitions and methods in Singularity Theory and Bifurcation Theory. In the third part of this paper, we list out the known results, and educe the research this paper will do. In the fourth part of this paper, a class of nonlinear BVP (boundary value problem) with bifurcation parameter is studied by using singularity theory. First, we use the Liapunov-Schmidt reduction transform the infinite-dimensional function space for the finite-dimensional space, and transform the dynamic branch for the static branch of the differential system to study. When the nonlinear term F is a bifurcation problem of finite-dimensional in the system, we prove that the bifurcation properties of F decides that of BVP by the method of singularity theory. The local bifurcation properties of equilibrium solution of the system follows from our results, including the information of the existence and the numbers of the bifurcation solutions.