Journal
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
Volume 24, Issue 4, Pages 713-759Publisher
SPRINGER
DOI: 10.1007/s10884-012-9271-4
Keywords
Symmetric Hopf bifurcation; Equivariant system; Differential-operator equation; Preisach hysteresis memory operator; Twisted equivariant degree
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Funding
- Alexander von Humboldt Foundation
- Science Foundation Ireland
- Russian Foundation for Basic Research [10-01-93112]
- Federal Programme 'Scientists of Innovative Russia' [2009-1.5-507-007]
- Universidad Nacional Autonoma de Mexico [PAPIIT IN-117511-3]
- Chutian Scholar Program
- China Three Gorges University in Yichang
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The standard approach to study symmetric Hopf bifurcation phenomenon is based on the usage of the equivariant singularity theory developed by M. Golubitsky et al. In this paper, we present the equivariant degree theory based method which is complementary to the equivariant singularity approach. Our method allows systematic study of symmetric Hopf bifurcation problems in non-smooth/non-generic equivariant settings. The exposition is focused on a network of eight identical van der Pol oscillators with hysteresis memory, which are coupled in a cube-like configuration leading to S (4)-equivariance. The hysteresis memory is the source of non-smoothness and of the presence of an infinite dimensional phase space without local linear structure. Symmetric properties and multiplicity of bifurcating branches of periodic solutions are discussed in the context showing a direct link between the physical properties and the equivariant topology underlying this problem.
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