Partially Coherent Twisted States in Two-Dimensional Arrays of Phase Oscillators: Interplay of Nonlocal Coupling and Heterogeneity

Abstract We consider an infinitely large two-dimensional array of nonlocally coupled phase oscillators with a Lorentzian distribution of natural frequencies. The simplest dynamical regimes supported by such a system are a completely disordered state (or uniform incoherence) and partially coherent twisted states, which are characterized by a linear increase in phase with distance along some direction. In the continuum limit, the coarse-grained dynamics of these regimes is described by an integro-differential equation derived from the Ott–Antonsen theory. In particular, partially coherent twisted states correspond to plane wave solutions of this equation. We perform a linear stability analysis of these waves, derive an explicit long-wave instability criterion for them, and analyze it. This allows us to find how stable twisted states with different wave vectors appear and change for increasing heterogeneity in the original oscillator system. The obtained analytical results are illustrated by their application to three specific types of radially symmetric nonlocal coupling.