Study On Equivariant Bifurcation Problems

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 Z₂ symmetry, in one parameter up to codimension 3; Golubitsky and Roberts studied the classification of degenerate Hopf bifurcation in two state variables with dihedron D₄ 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...