Frequency Spirals

TL;DR

This paper investigates localized rotating spiral patterns in coupled oscillator lattices near the phase-locking transition, revealing new dynamic structures called frequency spirals that are visible through instantaneous frequency analysis.

Contribution

It introduces and characterizes frequency spirals as a novel spatiotemporal pattern in Kuramoto lattices, combining analytical and numerical approaches.

Findings

Frequency spirals appear near the phase-locking transition.

They manifest as two-armed rotating spirals in instantaneous frequency.

Large lattice simulations show similar behavior to the simple prototype.

Abstract

We study the dynamics of coupled phase oscillators on a two-dimensional Kuramoto lattice with periodic boundary conditions. For coupling strengths just below the transition to global phase-locking, we find localized spatiotemporal patterns that we call "frequency spirals." These patterns cannot be seen under time averaging; they become visible only when we examine the spatial variation of the oscillators' instantaneous frequencies, where they manifest themselves as two-armed rotating spirals. In the more familiar phase representation, they appear as wobbly periodic patterns surrounding a phase vortex. Unlike the stationary phase vortices seen in magnetic spin systems, or the rotating spiral waves seen in reaction-diffusion systems, frequency spirals librate: the phases of the oscillators surrounding the central vortex move forward and then backward, executing a periodic motion with zero winding number. We construct the simplest frequency spiral and characterize its properties using analytical and numerical methods. Simulations show that frequency spirals in large lattices behave much like this simple prototype.