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Yes, there is. The keywords you're looking for are collective behavior and, in particular, synchronization in dynamical systems. And yes again: there must be some sort of coupling and, in a discrete model, the coupling between the individual oscillators will typically take the form of a synchronization network.

A recent (2015) review is Synchronization of chaotic systems, by Pecora and Carroll, and probably also worth mentioning are the book Dynamical System Synchronization by Luo and the highly-cited 2002 review The synchronization of chaotic systems by Boccaletti et al., but there's plenty of material on-line.

The last two questions are most interesting and unfortunately I can answer little more than to say that, yes, I think there might be a field theoretical approach to the problem, but all I could find in a quick search is the work of Ovchinnikov on Topological field theory of dynamical systems (paper II) (arxiv I, arxiv II).