Synchronization in non-uniform medium

In the previous oscillatory systems, the local dynamics was assumed to be the same everywhere in space and the synchronization of identical oscillatory regions was studied. In many cases the oscillatory medium is not uniform. In real chemical or biological systems this can be a consequence of non-uniform external conditions, like variation of temperature, or of illumination in a photosensitive reaction, 

The simplest form of a spatially non-uniform oscillatory reaction dynamics is obtained by assuming that the local dynamics is composed of identical limit cycles whose frequency is non-uniform in space. In this case the reaction term can be written in the form 

When the stirring rate is below a certain threshold the synchronized oscillation disappears and the mean concentration is almost constant apart from small irregular fluctuations that are independent of the stirring rate. In this regime the mixing is too slow to strongly couple the oscillatory dynamics of the fluid parcels. Snapshots of the 

The degree of synchronization can be characterized by the standard deviation in time of the spatially averaged concentration 

This order parameter measures the spatial coherence of the dynamics and can be normalized by dividing it with its value corresponding to the spatially uniform oscillatory system Ro = R(6 0), 0 R/Rq  

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