A note on Hopf bifurcation with dihedral group symmetry

• Type: Article in International Scientific Journal
• Date: 2006

Authors

dias, aps (Author) FCUP; paiva, rc (Author) Other

Journal

Title: Glasgow Mathematical Journal

Vol. 48

Pages: 41-51

ISSN: 0017-0895

Publisher: Cambridge University Press

Indexing

ISI Web of Knowledge - 4 Citations

Scopus - 4 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-004-PRG

DOI: 10.1017/s0017089505002855

Abstract (EN): We consider the standard action of the dihedral group D of order 2n on C. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on C circle plus C. Golubitsky and Stewart (Hopf bifurcation with dihedral group symmetry: Coupled nonlinear oscillators. In: Multiparameter Bifurcation Series, M. Golubitsky and J. Guckenheimer, eds., Contemporary Mathematics 46, Am. Math. Soc., Providence, R.I. 1986, 131-173) and van Gils and Valkering (Hopf bifurcation and symmetry: standing and travelling waves in a circular chain. Japan J. Appl. Math. 3, 207-222, 1986) prove the generic existence of three branches of periodic solutions, up to conjugacy, in systems of ordinary differential equations with D-n-symmetry, depending on one real parameter, that present Hopf bifurcation. These solutions are found by using the Equivariant Hopf Theorem. We prove that generically, when n not equal 4 and assuming Birkhoff normal form, these are the only branches of periodic solutions that bifurcate from the trivial solution.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 11