  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.
