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 $d_l=4$.