Synchronization unlocked: spirals, zetas, rings, and glasses

Abstract

Here, we study networks of coupled oscillators. Specifically, we identify phenomenology at or near a synchronization threshold in four distinct cases. First, we identify a novel spatiotemporal pattern in the two-dimensional Kuramoto lattice with periodic boundary conditions. This pattern appears as a two-armed rotating spiral in the spatial variation of the oscillators' instantaneous frequencies; hence the name ``frequency spirals.'' Second, we look at a large (but finite) number N of globally coupled oscillators in the special case where the natural frequencies are evenly spaced on a given interval. With these conditions, a leading order correction to the locking threshold is derivable, and scales according to N^{-3/2}. Thirdly, we do a case study on how topology can affect synchronization by comparing the locking threshold for a ring and chain of oscillators. Given identical initial phases and random natural frequencies, the ratio of locking thresholds is given upper and lower bounds which depend only on the shape of the coupling function. Finally, we examine a population of oscillators with random coupling strengths distributed across zero. A quarter century ago, a ``volcano transition'' was identified in such a model, but by using a particular coupling matrix construction, we present the first results analytically characterizing the transition point.