Collective synchronization in spatially extended systems of coupled oscillators with random frequencies

TL;DR

This study investigates the conditions under which spatially extended oscillators synchronize in phase and frequency, revealing critical dimensions and the impact of nonlinear effects on collective behavior.

Contribution

It provides analytical and numerical insights into the critical dimensions for phase and frequency synchronization in coupled oscillators with randomness.

Findings

Phase synchronization occurs only above four dimensions.

Frequency entrainment is possible in three dimensions.

Nonlinear effects lead to a disordered phase with runaway oscillators.

Abstract

We study collective behavior of locally coupled limit-cycle oscillators with random intrinsic frequencies, spatially extended over $d$-dimensional hypercubic lattices. Phase synchronization as well as frequency entrainment are explored analytically in the linear (strong-coupling) regime and numerically in the nonlinear (weak-coupling) regime. Our analysis shows that the oscillator phases are always desynchronized up to $d=4$, which implies the lower critical dimension $d_{l}^{P}=4$ for phase synchronization. On the other hand, the oscillators behave collectively in frequency (phase velocity) even in three dimensions ($d=3$), indicating that the lower critical dimension for frequency entrainment is $d_{l}^{F}=2$. Nonlinear effects due to periodic nature of limit-cycle oscillators are found to become significant in the weak-coupling regime: So-called {\em runaway oscillators} destroy the synchronized (ordered) phase and there emerges a fully random (disordered) phase. Critical behavior near the synchronization transition into the fully random phase is unveiled via numerical investigation. Collective behavior of globally-coupled oscillators is also examined and compared with that of locally coupled oscillators.