The Relative Finite Determinacy Of Function Germs And The Versality,Stability Of Multiparameter Equivariant Bifurcation Problems

Singularity theory came from H. M. Morse's critical point theory in the 1930s; in 1955 Whitney wrote the paper "On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane", this work makes it an independent branch, R. Thom, J. N. Mather, V. I. Arnold and M. Golubitsky and others made very important contributions in this regard. In our country under the leadership of Li Peixin researcher there are many scholars such as Li Yangcheng, Zhang Dunmu, Zhang Guobin, SunWeizhi, Zou Jiancheng, YuJianming, Jiang Guangfeng, Pei Donghe, as well as their students, their results enriched the singularity theory.In this paper several issues related to singularity theory are discussed, and the relative finite determinacy of function germs and the versality, stability of multiparameter equivariant bifurcation problems are two main issues of them.In 1978 P. F. S. Porto studied the relative function germs, which is the first paper to discuss relative function germs and relative map germs. From then on, there was a lot of discussion related to this issue, such as, in 1987 R. Bulajich, J. A. Gomez and L. Kushner's work "relative versality for map germs" published by the Bollettino UMI, in 1992 L. Kushner's paper about the relative finite determinacy on the algebraic set germs and his work in 2000 of finite relative determination and relative stability, in 2003 Tang Tieqiao and Li Yangcheng's work "R_r(S;n)-determinancy of function germs", in 2004 V. Grandjean's article "Infinite relative determinacy of smooth function germs with transverse isolated singularity and relative Lojasiewicz conditions" published by the journal of London Mathematical Society, then in 2004 Sun Weizhi, Chen Liang and Pei Donghe's paper "Strong Relative A_{S, T} Finite Determinacy of Map Germs and Relation with Relative Versal Unfolding" , in 2004 Shen Haiyan and Li Yangcheng's work "Relative A- determinacy of Relative Smooth map germs", in 2004 Li Yangcheng and Liang Qiongchu's paper "Versal Deformations of Relative Smooth Map-germs with Respect to Contact Equivalence", in 2007 Sun Weizhi, Gao Feng and Pei Donghe's work "versal deformations of relative map germs with K equivalence", and in 2006 Zhang Zhongfeng discussed the finite determicacy of group R(S;n) and R_r(S;n) in his Master thesis.There are a lot of related works with equivariant bifurcation problems at home and abroad, mainly to discuss the stability, unfoldings of equivariant bifurcation problems, the stability of equivariant bifurcation problems and their unfoldings, classification and recognition of them. When its state space is the same as the target space, the theoretical machinery from singularity theory are introduced by M. Golubitsky, I. Stewart and D. G. Schaeffer to study the equivariant bifurcation problems, and they got the equivariant universal unfolding theorem. Since then, many scholars continue to study this. Applying related methods and techniques in the theory of singularities of smooth map germs, A. L. Lavassani and Y. C. Lu studied the unfolding and stability of equivariant bifurcation problem. Particularly in our country, led by Professor Li Yangcheng, his students gave various versions of versality theorem. Professor Zhang Dunmu and his doctor Liu Hengxing discussed theΓ-equivariant (s , t)-equivalence relation andΓ-equivariant infinitesimally (r, s)-stability ofΓ-equivariant bifurcation problem.Based on singularity theory this paper discussed several problems as follows: (1) Versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups; (2) Infinitesimally stable unfolding of a class of equivariant bifurcation problems under equivariant left-right equivalent group; (3)Relative finite determinacy of smooth function germs; (4)Relative determinacy of deformations of function germs under the action of group; (5) the determinacy which discussed using DA algebra systems by A. L. Lavassani and Y. C. Lu, is analyzed without DA algebra systems in this paper. The first four parts are published in Acta Mathematica Scientia (English version, SCI), Natural Science Journal of Xiangtan University, Mathematica Applicata and Journal of Hunan University (Natural Sciences) respectively, and the last part is discussed carefully with doctor Cui Denglan and my tutor.The detailed contexts are as follows: Chapter 1 discusses versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups. For the unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry, the versal unfolding theorem with respect to left-right equivalence is obtained by using the related methods and techniques in the singularity theory of smooth map-germs. The corresponding results in the reference can be considered as its special cases. A relationship between the versal unfolding w. r. t. left-right equivalence and the versal deformation w. r. t. contact equivalence is established.Chapter 2 discusses infinitesimally stable unfolding of a class of equivariant bifurcation problems under equivariant left-right equivalent group. Applying the related methods and techniques in the singularity theory of smooth maps, infinitesimal stability of unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry is characterized. And the existence of infinitesimal stable unfolding of such a class of bifurcation problems is discussed.Chapter 3 studies relative finite determinacy of smooth function germs.Based on the works of P. F. S. Porto and A. du Plessis, it deals with the relative right determinacy of smooth function germs of n variables, which makes the same values on an algebraic set germ in Rⁿ. In this chapter the criteria on range of determinacy of such function germs are obtained. Some results in the chapter generalize or improve the corresponding ones in the reference.Chapter 4 discusses relative determinacy of deformations of function germs under the action of group. Based on the work of P. F. S. Porto and Andrew du Plessis, it deals with the relative right determinacy of deformations of function germs which are under the group R(S; n). In this chapter criteria on range of determinacy of deformations and a sufficient condition of the stability of deformations are obtained.Chapter 5 discusses the finite determinacy under contact equivalent group without DA algebra systems. Firstly, we studied carefully an article which is written by A. L. Lavassani and Y. C. Lu, and then considered that we need many basic algebra concepts to understand DA algebra systems, we tried to give another method to prove the finite determinacy without DA-algebra systems, lastly we obtained another method to prove lemma 6.2.2 in their paper, which means that we can prove the finite determinacy without DA-algebra systems. Considered that there has a special method in the proof, we put this in the last charpter.