Kardar-Parisi-Zhang fluctuations in the synchronization dynamics of limit-cycle oscillators

Abstract: The time-dependent process whereby one-dimensional systems of self-sustained oscillators synchronize is shown to display scale invariance in space and time, akin to that found in the dynamics of equilibrium critical phenomena. Remarkably, the process is largely independent of system details, sharing with a class of nonequilibrium surface kinetic roughening the universal scaling behavior of the Kardar-Parisi-Zhang equation with columnar noise, and featuring phase fluctuations that follow a Tracy-Widom probability distribution. This is revealed by a numerical exploration of rings of Stuart-Landau oscillators (the universal representation of an oscillating system close to a Hopf bifurcation) and rings of van der Pol oscillators, both paradigmatically supporting self-sustained oscillations. The critical behavior is very well defined for limit-cycle oscillations near bifurcation, and still dominates comparatively far from it. In particular, the Tracy-Widom fluctuation distribution seems to be an extremely robust feature of the synchronization process. The nonequilibrium criticality here described appears to transcend the details of the coupled dynamical systems that synchronize, making plausible its experimental observation.