Synchronization of oscillatory dynamics by mixing

Consider the continuum description of an oscillatory medium subject to advection and diffusion 

Here we also assume that the reaction term does not depend explicitly on the spatial coordinate, therefore the dynamics of the medium is uniform in space. It is easy to see that the spatially uniform time-periodic oscillation is a trivial solution of the full reaction-diffusion-advection system, so the question is whether this uniform solution is stable to small non-uniform perturbations and more generally, if there are any persistent spatially non-uniform solutions in which the spatial structure does not decay in time. 

The qualitative behavior of the system depends on various characteristics of the velocity field. The simplest case is when there is uniform chaotic mixing in the flow over the whole domain, so that there are no transport barriers and the characteristic length-scale of the velocity field is comparable to the size of the domain. This problem has been studied by Kiss et al. (2004) using a model of the chlorine-iodine-malonic acid (CDIMA) reaction (Sect. 3.1.4) described by 

The process of synchronization of the local oscillations can be described by following the evolution of the standard deviation in space of the concentration field as a function of time (Fig. 8.1) 

This can be shown more quantitatively for the case of a spatially localized inhomogeneity, where the stability of the uniform 

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