Collective phase synchronization in locally coupled limit-cycle oscillators

We study collective behavior of locally coupled limit-cycle oscillators with scattered intrinsic frequencies on [Formula presented]-dimensional lattices. A linear analysis shows that the system should always be desynchronized up to [Formula presented]. On the other hand, numerical investigation for [Formula presented] and [Formula presented] reveals the emergence of the synchronized (ordered) phase via a continuous transition from the fully random desynchronized phase. This demonstrates that the lower critical dimension for the phase synchronization in this system is [Formula presented].