Universal behavior in the dynamics of synchronizing oscillators and surface growth phenomena

Ricardo Gutiérrez

Mathematics Department, Carlos III University of Madrid

Avenida de la Universidad 30, 28911, Leganés, Spain

The study of synchronous dynamics has traditionally focused on the identification of threshold parameter values for the transition to synchronization, and on the nature of such transition from a static point of view. The dynamical process whereby systems of self-sustained oscillators eventually synchronize at long times, however, has been much less studied. While one might reasonably expect such a process to be strongly system-dependent, we have recently shown that indeed the opposite is the case, as it contains robust universal features that have been previously studied in the context of nonequilibrium critical dynamics. This behavior can be explained from a mathematical connection existing between spatially discrete synchronization models and the continuum equations of surface growth processes. By means of detailed numerical studies of one-dimensional systems of phase oscillators and several limit-cycle oscillators in the presence of quenched disorder [1,2] and time-dependent noise [3], we provide strong evidence confirming that the synchronization process in these systems is characterized by forms of generic scale invariance associated with the universality classes of kinetically rough interfaces, such as the Kardar-Parisi-Zhang (KPZ) universality class [3], and that of the KPZ equation with columnar disorder [1,2]. Moreover, the phase fluctuations around the average growth follow a ubiquitous Tracy-Widom probability distribution, which is frequently associated with the KPZ nonlinearity. Synchronization and surface growth processes appear to be much more closely related than previously anticipated, which opens up many avenues for future research, including the experimental observation of the critical dynamics of the synchronization process. Work done in collaboration with Rodolfo Cuerno.

[1] R. Gutiérrez and R. Cuerno, Nonequilibrium criticality driven by Kardar-Parisi-Zhang fluctuations in the synchronization of oscillator lattices, Phys. Rev. Research 5, 023047 (2023). 

[2] R. Gutiérrez and R. Cuerno, Kardar-Parisi-Zhang fluctuations in the synchronization dynamics of limit-cycle oscillators, arXiv:2311.13253 (2023)

[3] R. Gutiérrez and R. Cuerno, Kardar-Parisi-Zhang universality class in the synchronization of oscillator lattices with time-dependent noise, (in preparation).