Bifurcation Behavior of D _n -Equivariant Systems

Abstract

The general framework of the group-theoretic bifurcation theory has been presented in the previous chapter. In this chapter, it is applied to the simplest group, the dihedral group D _n , which represents the symmetry of the regular n-sided polygon. This is based on standard studies of the bifurcation behavior of a D _n -equivariant system (Sattinger, 1979, 1983 [159], [161]; Fujii, Mimura, and Nishiura, 1982 [51]; Golubitsky, Stewart, and Schaeffer, 1988 [58]; Healey, 1985, 1988 [65], [66]; Delinitz and Werner, 1989 [37]; Ikeda, Murota, and Fujii, 1991 [91]). A few remarks are given about a system equivariant to a cyclic group C _n , which has a partial symmetry of D _n . While simple critical points and double bifurcation points appear inherently in D _n - and C _n -equivariant systems, emphasis naturally is to be placed on the double bifurcation points, because most of the results in Chapters 2 and 3 are applicable to the simple critical points of these systems.