The Classification And Recognition Of Bifurcation Problem With Trivial Solution

This paper mainly studies the classification and the recognition of bifurcation problem with trivial solution under the action of(?)-equivalence group.M.Golubitsky and D.Schaef-fer applied the methods of singularity theory and group theory into the research of bifurcation theory.In this paper,we use the above method to study the bifurcation problem with trivial solution that is acted by the(?)-equivalence group.We study the finite determination of the bifurcation problem.Then we give the sufficiency theorem to decide that two bifurcation problems are(?)-equivalent and we give classification theorem of bifurcation problem with codimension up to 3.We define the smallest intrinsic submodule containing the bifurcation problem and the high order terms of the bifurcation problem in order to recognize them.We analyse the properties and formulas of them.Thesis is an application of the above methods and also a complement to the bifurcation theory.In this paper,we define the bifurcation problem with trivial solution and the group(?)which is formed by(?)-equivalence.Then we give the orbit tangent space of bifurcation problem under the action of the group(?).Since the orbit tangent space is not an ideal or module of the ring ?x,?.It brings difficulty on deciding whether bifurcation problem has finite codimension.By Damon's method,we get the necessary and sufficient condition that bifurcation problem has finite codimension.This is the first main result in this thesis.Then we introduce the invariable set under the action of(?)-equivalence the intrinsic submodule and give the property of it.We study the sufficient condition to decide the two bifurcation problem are(?)-equivalent on the neighbourhood of origin,then we generalize it on interval[0,1].Next,we analyse the characteristics of the bifurcation problem with trivial solution.We prove the classification theorem about bifurcation problem with codimension up to 3,this is the second main result.Last,we study the recognition solution of normal form.We define the smallest intrinsic submodule(?)(h),analyse its property and the calculate methods of lower terms and inter-mediate order terms.We give the definition of high order terms(?)(h)of the bifurcation problem.We also analyse the property and the formula of the high order terms.By M.Gol-ubitsky's methods,we can not get the formula of the high order terms of bifurcation problem with trivial solution.Because we can not solve the three variables from a equation set with two equations.So we consider a normal subgroup(?)of(?)and define the orbit tangent space under the action of this normal subgroup(?).The high order term under the action of(?)is denoted as(?)(h,(?)).We analyse the relation of high order term(?)(h)and the high order term(?)(h,(?)),then we get the formula of(?)(h).Thus we get the third main result of this paper.