等变分歧问题研究
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摘要
自从上世纪70年代，Golubitsky M．和Schaeffer D．G．引入奇点理论和群论方法研究分歧问题的思想，分歧理论得到迅猛发展。研究分歧问题主要是如何将光滑映射奇点理论中的相关概念和技巧适当地引到分歧问题中来。这些研究主要有以下两个方面的问题：
第一，分歧问题的分类与识别。这是一个非常有意义但又很棘手的问题。它研究分歧问题在等价意义下有几类，它们的标准形式是什么。研究分歧问题在什么条件下等价于给定的标准形式。为此必须寻找这些标准形在等价群作用下的轨道特征。借助于奇点理论中的有限决定性可以将无限维识别问题转化为有限维的情形来处理。等价群模去高阶项将以Lie群方式作用在一个有限维空间上，这样将轨道描述成由一些这样的映射芽组成，它们的Taylor系数满足有限多个多项式方程或不等式，这正是识别问题的解。到目前为止，只解决了几类分歧问题在低余维条件下的分类与识别问题。如Keyfitz给出了只有一个状态变量且不带对称性、余维数不大于7的分歧问题的分类；Golubitsky和Schaeffer得到了单状态变量，关于Z_2对称，单参数，余维数不超过3的分歧问题的分类；Golubitsky和Roberts研究了两状态变量关于正交群O(2)对称的退化Hopf分歧的分类以及两状态变量关于二面体群D_4对称、单参数、拓扑余维不超过2的分歧问题的分类；Melbourne得到了三状态变量关于八面体群对称，单参数，拓扑余维不大于1的分歧问题的分类。需要指出的是以上研究均没有考虑分歧参数的对称性，Futer，Sitta和Stewart的工作虽然考虑了分歧参数的对称性，但仅限于分歧参数与状态变量具有相同的对称性，他们得到了状态变量与分歧参数均关于二面体群D_4对称，拓扑余维数不大于1的分歧问题的分类。高守平和李养成则讨论状态变量和分歧参数均具有对称性且对称性可以不同的分歧问题，并给出了状态变量关于二面体群D_4对称，分歧参数关于S~1对称，拓扑余维不超过1的分歧问题的分类。本文第一章讨论两个状态变量关于二面体群D_3对称，两个分歧参数关于O(2)对称的分歧问题，给出了该类分歧问题在非退化条件q(0)≠0下所有情形的分类与相应的识别条
Ever since the 70s of last century Golubitsky M. and Schaeffer D. G have introduced the idea of applying the methods and techniques of singularity and group theory to the study of bifurcation problems, the bifurcation theory has rapidly developed . The study of bifurcation problems is mainly about how to imply the related concepts and techniques in singularity theory of smooth map germs to bifurcation problems. It includes the following two major aspects:
Firstly, the classification and recognition of bifurcation problems. It is a very meaningful but rather difficult subject discussing how many classes bifurcation problems have under some equivalence, what their normal forms are, when a bifurcation problem is equivalent to a given normal form. So we must find the orbital characteristics of these normal forms under the action of some equivalent group. The use of the finite determinacy in singularity theory can transfer infinitely dimensional recognition to finitely dimensional recognition. Modulo high order terms the equivalence group acts as a Lie group on a finitely dimensional space, 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. So far the classifications and recognitions of only a few types of bifurcation problems under the condition of low codimensions have beem completed. For example, Keyfitz made the classification of the bifurcations in one state variable, without symmetry up to codimension 7; Golubitsky and Schaeffer obtained the classification of the bifurcation problems in one state variable with Z_2 symmetry, in one parameter up to codimension 3; Golubitsky and Roberts studied the classification of degenerate Hopf bifurcation in two state variables with dihedron D_4 symmetry, in one parameter up to topological codimension 2; Melbourne obtained the classification of bifurcations in three state variables with octahedral symmetry, in one parameter, up to
第一，分歧问题的分类与识别。这是一个非常有意义但又很棘手的问题。它研究分歧问题在等价意义下有几类，它们的标准形式是什么。研究分歧问题在什么条件下等价于给定的标准形式。为此必须寻找这些标准形在等价群作用下的轨道特征。借助于奇点理论中的有限决定性可以将无限维识别问题转化为有限维的情形来处理。等价群模去高阶项将以Lie群方式作用在一个有限维空间上，这样将轨道描述成由一些这样的映射芽组成，它们的Taylor系数满足有限多个多项式方程或不等式，这正是识别问题的解。到目前为止，只解决了几类分歧问题在低余维条件下的分类与识别问题。如Keyfitz给出了只有一个状态变量且不带对称性、余维数不大于7的分歧问题的分类；Golubitsky和Schaeffer得到了单状态变量，关于Z_2对称，单参数，余维数不超过3的分歧问题的分类；Golubitsky和Roberts研究了两状态变量关于正交群O(2)对称的退化Hopf分歧的分类以及两状态变量关于二面体群D_4对称、单参数、拓扑余维不超过2的分歧问题的分类；Melbourne得到了三状态变量关于八面体群对称，单参数，拓扑余维不大于1的分歧问题的分类。需要指出的是以上研究均没有考虑分歧参数的对称性，Futer，Sitta和Stewart的工作虽然考虑了分歧参数的对称性，但仅限于分歧参数与状态变量具有相同的对称性，他们得到了状态变量与分歧参数均关于二面体群D_4对称，拓扑余维数不大于1的分歧问题的分类。高守平和李养成则讨论状态变量和分歧参数均具有对称性且对称性可以不同的分歧问题，并给出了状态变量关于二面体群D_4对称，分歧参数关于S~1对称，拓扑余维不超过1的分歧问题的分类。本文第一章讨论两个状态变量关于二面体群D_3对称，两个分歧参数关于O(2)对称的分歧问题，给出了该类分歧问题在非退化条件q(0)≠0下所有情形的分类与相应的识别条
Ever since the 70s of last century Golubitsky M. and Schaeffer D. G have introduced the idea of applying the methods and techniques of singularity and group theory to the study of bifurcation problems, the bifurcation theory has rapidly developed . The study of bifurcation problems is mainly about how to imply the related concepts and techniques in singularity theory of smooth map germs to bifurcation problems. It includes the following two major aspects:
Firstly, the classification and recognition of bifurcation problems. It is a very meaningful but rather difficult subject discussing how many classes bifurcation problems have under some equivalence, what their normal forms are, when a bifurcation problem is equivalent to a given normal form. So we must find the orbital characteristics of these normal forms under the action of some equivalent group. The use of the finite determinacy in singularity theory can transfer infinitely dimensional recognition to finitely dimensional recognition. Modulo high order terms the equivalence group acts as a Lie group on a finitely dimensional space, 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. So far the classifications and recognitions of only a few types of bifurcation problems under the condition of low codimensions have beem completed. For example, Keyfitz made the classification of the bifurcations in one state variable, without symmetry up to codimension 7; Golubitsky and Schaeffer obtained the classification of the bifurcation problems in one state variable with Z_2 symmetry, in one parameter up to codimension 3; Golubitsky and Roberts studied the classification of degenerate Hopf bifurcation in two state variables with dihedron D_4 symmetry, in one parameter up to topological codimension 2; Melbourne obtained the classification of bifurcations in three state variables with octahedral symmetry, in one parameter, up to
引文
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[3] Golubitsky M., Roberts. M. A classification of degenerate Hopf bifurcation with O(2) symmetry. J. Diff. Equa. 1987, 69: 216～264.
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[7] Dangelamyr G., Armbruster D. Classfication of Z_2-equivariant imperfect bifurcations with corank 2. Proc. London Math. Soc. 1983, 46 (3): 517～547.
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[9] Gao S. P., Li Y. C. Classification of (D_4, S~1) - equivariant bifurcation problems up to topological codimension 2. Science in China, Ser. A, 2003, 46 (6): 863～871.
[10] Golubitsky M., Stewart I., Schaeffer D. G. Singularities and groups in bifurcation theory. Vol 2, New York: Springer-Verlag, 1988.
[11] Damon J. The unfolding and determinacy theorems for subgroups of (?) and (?). Memoirs of the Amer. Math. Soc., 1984, 60 (506): 1-88.
[12] Li Y. C. The recognition of eptdvariant bifurcation problems. Science in China, Ser. A, 1996, 39 (6): 604～612.
[13] Melbourne I. The recognition problem for equivariant singularties. Nonlinearity, 1988, 1: 215～240.
[14] Li Y. C. Singularity theory of smooth mappings. Beijing: Science Press, 2002 (in Chinese).
[15] 郭瑞芝，李养成．余维数不大于3的(D_3，O (2))-等变分歧问题的分 类．数学学报，2006，49(2)：255～264．
[16] Wall C T C. Finity Determinacy of smooth map-germs. Bull. London Math. Soc., 1981, 13: 481～539.
[17] Gao Shouping, Li Yangcheng. The unfolding of equivariant bifurcation problems with parameters symmetry. Acta Mathematica Scientia. 2004, 24B (4): 623～632.
[18] 张锦炎，冯贝叶．常微分方程几何理论与分支问题．北京：北京大学出版社，2000．
[19] 邹建成．on Singularities in bifurcation theory．[博士学位论文]．北京：中国科学院数学研究所，1996．
[20] Dangelmayr G., Stewart I. Classification and Unfolding of Sequential Bifurcation. SIAM J. Math. Anal.
[21] Gaffney T. New methods in classification theory of bifurcation problem. Contemporary Mathematics. 1986, (56): 97～116.
[22] Golubitsky M., Guillemin V. Contact equivalence for Lagrange manifolds. Adv. In Math. 1975, (15): 375～387.
[23] Golubitsky M, Guillemin V. Stable mappings and their singularities. New York: Springer-Verlag, 1973.
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