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Hopf bifurcation in symmetric networks of coupled oscillators with hysteresis. (English) Zbl 1264.34093
Summary: The standard approach to the study of the 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 a method based on equivariant degree theory which is complementary to the equivariant singularity approach. Our method allows for a 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. The 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.

MSC:
34C55 Hysteresis for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37G40 Dynamical aspects of symmetries, equivariant bifurcation theory
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