On Unfolding,Stability And Recognition Of Multiparameter Equivariant Bifurcation Problems

In 1979, Golubitsky M. and Schaeffer D. G introduced the idea of applying the methods and techniques of singularity and group theory to the study of bifurcation problems. The tools used to the study of bifurcation problems come mainly from related techniques in the theory of singularities of smooth map germs. It mainly includes the following aspects:Firstly, the unfoldings of bifurcation problems. It studies parameterized families of perturbations of the given bifurcation problems. If there exists a versal unfolding for a bifurcation problem f, then every unfolding which is perturbation off can be factored through the versal unfolding. So the study of versal unfoldings of bifurcation problems is very interesting. For example, Golubitsky, Stewart gives the universal unfolding theorem of one parameter equivariant bifurcation problems when the state space of a bifurcation problem is the same as its target space. Afterwards many scholars continue to study this problem, especially in China, Proffessor Li Yangcheng and his students establish kinds of versions of the versal unfolding theory.Secondly, stability of bifurcation problems and their unfoldings. The discussion of various stabilities of smooth map-germs is an important part of the study of singularity theory. By using the techniques from singularity theory, Lavassani and Lu discuss the stability of equivariant bifurcation problems and their unfoldings; Liu Hengxing,Professor Zhang Dunmu discuss the (r,s)-stability of unfoldings of bifurcation problems.Thirdly, the classification and recognition of bifurcation problems. To know precisely when a bifurcation problem is equivalent to a given normal form, we must find the characteristics of the orbit of the normal forms under the action of some equivalent group. This problem can often be reduced to one of finite dimensions by using a key idea the finite determinacy in singularity theory. The equivalent group acts as a Lie group on a finitely dimensional space by modulo high order terms, thus its orbits can be characterized as comprising those germs whose Taylor coefficients satisfy a finite number of polynomial constraints in the form of equalities and inequalities. This characterization is just the solution to the recognition problems. In the study of the recognition of bifurcation broblems, Gaffney establishes D(Γ) -equivalence theory by introducing unipotent algebraic group and nilpotent Lie algebras. Melbourne study equivariant bifurcation problems with one bifurcation parameter and establishes U(Γ) -equivalence theory. Li Yangcheng establishes U(Γ) -equivalence of multiparameter equivariant bifurcation problems more generally. The classification of bifurcation problems is a very meaningful but rather difficult subject. It discusses how many classes bifurcation problems have under some equivalence, and what their normal forms are. So far the recognitions and classifications of only a few types of bifurcation problems under the condition of low codimensions havebeem completed.Generally, the symmetry of bifurcation parameters is not considered in the study of equivariant bifurcation problems. Even if it has been considered, the study is limitted to the case of the parameter space has the same symmetry as the state space, and all state variables of the bifurcation problem are treated equally without any distinction. In this paper, we study the steady-state multiparameter equivariant bifurcation problems with symmetry on bifurcation parameters, which may be different from that of state variables, and the state variables are divided into two types. The first type can vary independently, while the other type depends on the previous one when varying.In chapter 1, the unfolding of such bifurcation problems under the action of contact equivalent group was discussed. The versal unfolding theorem obtained in this paper by using DA-algebras tools introduced by Damon not only includes many corresponding results but also cancels the stronger demand on the codimension of a bifurcation Problem in some of relevant references to establish more general unfolding theory.In chapter 2, we continued to study the stabilities of such equivariant bifurcation problems and their unfoldings under the action of contact equivalent group. The equivalence of the stability and infinite stability of such equivariant bifurcation problems is obtained. Transversality condition is used to characterize the stability of such equivariant bifurcation problems.In chapter 3, equivariant bifurcation problem with symmetry both on the state variables and the bifurcation parameters was discussed. Two recognition conditions on equivariant bifurcation problems with D6 symmetry on the state variables and Z₂ symmetry on the bifurcation parameters was obtained. The results obtained here was the base of the classification of such equivariant bifurcation problems.In chapter 4, the stability of unfolding of equivariant bifurcation problems with parameters symmetry under the action of left-right equivalent group was discussed. The infinitesimal stability of the unfolding of such equivariant bifurcation problems was characterized.In chapter 5, the unfolding of equivariant bifurcation problems with parameters symmetry under the action of a subgroup of left-right equivalent group was discussed.