Generalized splay states in phase oscillator networks. (English) Zbl 1468.34042
Summary: Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we introduce generalized \(m\)-splay states constituting a special subclass of phase-locked states with vanishing \(m\) th order parameter. Such states typically manifest incoherent dynamics, and they often create high-dimensional families of solutions (splay manifolds). For a general class of phase oscillator networks, we provide explicit linear stability conditions for splay states and exemplify our results with the well-known Kuramoto-Sakaguchi model. Importantly, our stability conditions are expressed in terms of just a few observables such as the order parameter or the trace of the Jacobian. As a result, these conditions are simple and applicable to networks of arbitrary size. We generalize our findings to phase oscillators with inertia and adaptively coupled phase oscillator models.
©2021 American Institute of Physics
©2021 American Institute of Physics
MSC:
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
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