The Recognition Of Two Dimensional Bifurcation Problem Under T-Equivalent

The research on the classification and recognition of bifurcation problems is quite meaningful.It consists of the following several aspects:how many types do bifurcation problems have under the equivalent meaning,what are their normal forms ,and under what conditions the bifurcation problem is equivalent to the standard form.So we must search for the orbital characteristics of these normal forms under the action of the equivalence group.Using finite determinacy of singularity theory,we can turn an infinite dimensional recognition problem to a finite dimensional one .The orbit can be described as those map germs whose Taylor-coefficients satisfy finitely many polynomial equations or inequalities, these just are the solutions of the recognition problem.Firstly,in this paper,the concept of 2-dimensional bifurcation problem with equilibria is given and a subgroup of the contact equivalent group, called T-equivalent group,is introduced in this paper.The bifurcation problems with equilibria are preserved. The concepts of T-equivalent relations and corresponding orbits are constructed. According to the definition of the tangent space of a manifold with finite dimension,the tangent space of a orbit under the T-equivalent group is given.Secondly,the algebraic construction and characteristics of the tangent space of a bifurcation problem are studied and analyzed,by using of the finitely determined technique, Nakayama lemma and the related knowledge in algebra.It is proved that the bifurcation problems with the same tangent space are equivalent under a certain condition. Finally,the normal form and the corresponding recognition condition of bifurcation problems are obtained under some conditions.