Influence of coupling symmetries and noise on the critical dynamics of synchronizing oscillator lattices

Abstract

Recent work has shown that the synchronization process in lattices of self-sustained (phase and limit-cycle) oscillators displays universal scale-invariant behavior previously studied in the physics of surface kinetic roughening. The type of dynamic scaling ansatz which is verified depends on the randomness that occurs in the system, whether it is columnar disorder (quenched noise given by the random assignment of natural frequencies), leading to anomalous scaling, or else time-dependent noise, inducing the more standard Family-Vicsek dynamic scaling ansatz, as in equilibrium critical dynamics. The specific universality class also depends on the coupling function: for a sine function (as in the celebrated Kuramoto model) the critical behavior is that of the Edwards-Wilkinson equation for the corresponding type of randomness, with Gaussian fluctuations around the average growth. In all the other cases investigated, Tracy–Widom fluctuations ensue, associated with the celebrated Kardar–Parisi–Zhang equation for rough interfaces. Two questions remain to be addressed in order to complete the picture, however: (1) Is the atypical scaling displayed by the sine coupling preserved if other coupling functions satisfying the same (odd) symmetry are employed (as suggested by continuum approximations and symmetry arguments)? and (2) how does the competition between both types of randomness (which are expected to coexist in experimental settings) affect the nonequilibrium behavior? We address the latter question by numerically characterizing the crossover between thermal-noise and columnar-disorder criticality, and the former by providing evidence confirming that it is the symmetry of the coupling function that sets apart the sine coupling, among other odd-symmetric couplings, due to the absence of Kardar–Parisi–Zhang fluctuations.