The Classification Of Bifurcation Problems Under T-Equivalent Group

Classification problem is an important subject of bifurcation theory.In application, many models are bifurcation problems with trivial solution.In this paper,we mainly study the classification of bifurcation problems with trivial solution with codimention not more than 3 under t- equivalent group.The content of this paper will be mainly divided into four parts.In the part of introduction, we briefly introduce the development of singularity theory. Next, we introduce the application of singularity theory in bifurcation theory and the development of the classification of bifurcation problems.In the second part, we introduce some definitions and methods of singularity theory. Then, we introduce some definitions of bifurcation theory , for example, bifurcation problem,tangent space,the co-dimension,etc .At last, we go over the classification of bifurcation problems with codimension not more than 3 under strong equivalent group.In the third part, we first introduce the definitions of bifurcation problems with trivial solution,the orbit tangent space and the co-dimension . Then, we give one of the main results of this paper, i.e., the sufficient condition to decide two bifurcation problems with trivial solution to be t-equivalent. At last, we discuss the finite t-determinacy of bifurcation problem with trivial solution, and give the second result of this paper, i.e., the sufficient condition for smooth function germs g∈ε_x,λ·{ x}and k - jet(k∈N ) of g to be t- equivalent.In the fourth part of this paper, we give the third result of this paper,i.e.,the classification theorem of bifurcation problem with trivial solution with codimension not more than 3 under t-equivalent group.Then we have given the detailed proof of the classification theorem.