Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations

Zalman Balanov; Meymanat Farzamirad; Wieslaw Krawcewicz; Haibo Ruan

doi:10.3934/dcds.2006.16.923

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2006, Volume 16, Issue 4: 923-960. Doi: 10.3934/dcds.2006.16.923

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Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations

1.
Department of Mathematics and Computer Sciences, Netanya Academic College, Netanya 42365
2.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1

Received: May 31, 2005

Revised: July 31, 2006

Published: September 2006

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Abstract

In this paper we apply the equivariant degree method to the Hopf bifurcation problem for a system of symmetric functional differential equations. Local Hopf bifurcation is classified by means of an equivariant topological invariant based on the symmetric properties of the characteristic operator. As examples, symmetric configurations of identical oscillators, with dihedral, tetrahedral, octahedral, and icosahedral symmetries, are analyzed.

Keywords:

Equivariant degree,
symmetric systems of FDEs,
Hopf bifurcation with symmetries,
basic maps,
isotypical crossing numbers,
configurations of identical oscillators,
dihedral,
tetrahedral,
octahedral and icosahedral symmetries.

Mathematics Subject Classification: Primary: 34C25; Secondary: 47J05.

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