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Local bifurcation in symmetric coupled cell networks: linear theory. (English) Zbl 1112.34025

The subject of the paper are coupled cell networks composed of coupled ordinary differential equations

dx j /dt=f j (x 1 ,,x N ),x j l ,j=1,,N·

Under the assumption that the system possesses a finite symmetry group Γ, the linearized vector field is investigated at a fully symmetric equilibrium.

It is shown that in the case when the cells are active, i.e., f j depends on x j , and Γ is Abelian, the network structure does not influence codimension-one local bifurcations. Beyond this context, when Γ is not Abelian and cells are passive (f j does not depend on x j ) anomalies in higher-dimensional local bifurcations, due to the network structure may arise.

34C23Bifurcation (ODE)
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C14Symmetries, invariants (ODE)