Invariants, equivariants and characters in symmetric bifurcation theory

• Type: Article in International Scientific Journal
• Date: 2008

Authors

antoneli, f (Author) Other; dias, aps (Author) FCUP; matthews, pc (Author) Other

Journal

Title: Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Vol. 138

Pages: 477-512

ISSN: 0308-2105

Publisher: Cambridge University Press

Indexing

ISI Web of Knowledge - 6 Citations

Scopus - 7 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-004-4MB

DOI: 10.1017/s0308210506001119

Abstract (EN): 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 ail equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is ail 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 sonic 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'. We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group O(3).

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 36