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Networks of coupled phase oscillators play an important role in the analysisof emergent collective phenomena. In this article, we introduce generalized$m$-splay states constituting a special subclass of phase-locked states withvanishing $m$th order parameter. Such states typically manifest incoherentdynamics, and they often create high-dimensional families of solutions (splaymanifolds). For a general class of phase oscillator networks, we provideexplicit linear stability conditions for splay states and exemplify our resultswith the well-known Kuramoto-Sakaguchi model. Importantly, our stabilityconditions are expressed in terms of just a few observables such as the orderparameter or the trace of the Jacobian. As a result, these conditions aresimple and applicable to networks of arbitrary size. We generalize our findingsto phase oscillators with inertia and adaptively coupled phase oscillatormodels.

Generalized splay states in phase oscillator networks