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We study collective behavior of locally coupled limit-cycle oscillators withrandom intrinsic frequencies, spatially extended over $d$-dimensionalhypercubic lattices. Phase synchronization as well as frequency entrainment areexplored analytically in the linear (strong-coupling) regime and numerically inthe nonlinear (weak-coupling) regime. Our analysis shows that the oscillatorphases are always desynchronized up to $d=4$, which implies the lower criticaldimension $d_{l}^{P}=4$ for phase synchronization. On the other hand, theoscillators behave collectively in frequency (phase velocity) even in threedimensions ($d=3$), indicating that the lower critical dimension for frequencyentrainment is $d_{l}^{F}=2$. Nonlinear effects due to periodic nature oflimit-cycle oscillators are found to become significant in the weak-couplingregime: So-called {\em runaway oscillators} destroy the synchronized (ordered)phase and there emerges a fully random (disordered) phase. Critical behaviornear the synchronization transition into the fully random phase is unveiled vianumerical investigation. Collective behavior of globally-coupled oscillators isalso examined and compared with that of locally coupled oscillators.Comment: 18 pages, 18 figure

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