Two-cluster bifurcations in systems of globally pulse-coupled oscillators. (English) Zbl 1252.34042
The paper investigates the emergence and stability of in-phase synchronized and two-cluster states in a system of globally pulse-coupled identical phase oscillators. The coupling among oscillators is based on the phase response curve (PRC) which reflects a phase shift in oscillatory dynamics of an individual oscillator in response to a precisely timed perturbation. PRC is widely used for the modeling of biological and neural oscillators. For the dynamics of the network reduced to a two-cluster subspace, a one-dimensional map is obtained, and the stability conditions for the above states are derived in relation to the properties of the PRC. It is also shown that a two-cluster state can emerge from the one-cluster state via a pitchfork bifurcation. A special case of smooth PRCs vanishing at the onset of spike together with their first derivatives is considered in detail. The obtained analytical results are illustrated on numerical example given by a one-parameter family of PRCs of type I with positive coupling constant modeling an excitatory coupling in neural networks. For such large networks a completely synchronous state appears to be locally unstable, however, it forms an attractive homoclinic loop which supports an intermittent synchronized dynamics. As the shape of PRC changes, numerous two-cluster states bifurcate from the in-phase synchronized state. The corresponding parameter region of multiple co-existing stable two-cluster states is found in the space of PRC parameter and cluster configuration. The observed phenomena are also shown to be robust with respect to small parameter perturbations.
Reviewer: Oleksandr Popovych (Jülich)
MSC:
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34D06 | Synchronization of solutions to ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34D10 | Perturbations of ordinary differential equations |
Keywords:
Phase oscillators; pulse coupling; synchronization; clusters; phase response curve; bifurcationReferences:
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