Collective phase synchronization in locally coupled limit-cycle oscillators

We study collective behavior of locally coupled limit-cycle oscillators with scattered intrinsic frequencies on d-dimensional lattices. A linear analysis shows that the system should always be desynchronized up to d=4. On the other hand, numerical investigation for d=5 and d=6 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 d(l)=4.