The Research Of Finite Codimension Of Equivariant Gradient Map Germs

In the bifurcation theory,an important research topic is the study of variable bifurcation problem under the equivalent of the standard forms.And then,researcher gives a classification and recognition.However,classifications and recognitions need to ensure the finiteness of its codimension,so it is necessary to discuss the finiteness of its codimension for equivariant gradient map germs.Besides,equivariant gradient map germ comes from invariant function germ.Therefore,in this paper,we want to find the relationship between invariant function germs and their gradient map germs and study the finite codimension in equivariant gradient bifurcation problem by using singularity theory and group theory.This paper consists of four chapters.In the first chapter: We discuss the development trends and research meaning and make a brief summary of reference date.In the second chapter: We introduce some basic concepts and some basic conclusion.In the third chapter: We mainly introduce various relationships between invariant function germs and equivariant gradient map germs.In the fourth chapter: We first testify that the gradient codimension is an invariant of the contact equivalence with a bifurcation parameter and a symmetry group and prove the equivalence between finite codimensionand finite gradient codimension for equivariant gradient bifurcation problem.