Invariants, Equivariants and Characters in Symmetric Bifurcation Theory

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory, and in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as “Molien functions”. In this work we obtain formulas for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters and we show that they are effectively computable in several concrete examples. This information allows to draw some predictions about the structure of the bifurcations. For example, by comparing the number of equivariants with the number of invariants of one higher degree, it can be checked immediately whether the dynamics is variational (gradientlike). Mathematics Subject Classification: 37G40, 37C80, 13A50 ∗Department of Applied Mathematics, University of São Paulo, São Paulo, SP 05508-090, Brazil. †Centro de Matemática and Dep. Matemática Pura, Universidade do Porto, Porto, 4169-007, Portugal ‡School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom