Collective behavior in a chain of van der Pol oscillators with power-law coupling

We consider the synchronization properties of a one-dimensional chain of coupled van der Pol oscillators. The uncoupled oscillators have an attracting limit-cycle with a given normal mode frequency. We introduce a given level of randomness in the normal mode distribution of the oscillators and study the conditions under which the chain synchronizes. The coupling depends on the distance along the lattice in a power-law fashion. There is a frequency synchronization transition as we pass continuously from a global to a local coupling. We observe phase and lag synchronization transition for other coupling regimes.