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Amplitude equations and fast transition to chaos in rings of coupled oscillators

We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of Ginzburg-Landau type, which describes the destabilization of a uniform stationary state in a ring with a large number of nodes. Applying these results to unidirectionally coupled Duffing oscillators, we explain the phenomenon of a fast transition to chaos, which has been numerically observed in such systems. More specifically, the transition to chaos occurs on an interval of a generic control parameter that scales as the inverse square of the size of the ring, i.e. for sufficiently large system, we observe practically an immediate transition to chaos.