Spatial pattern formation

Krischer, K. Principles of Temporal and Spatial Pattern Formation in Electrochemical Systems 32... 

An alternative way of portraying the pattern formation behaviour in systems of the sort under consideration here is to delineate the regions in chemical parameter space (the h k plane) over which the uniform state is unstable to non-uniform perturbations. We have already seen in chapter 4, and in Fig. 10.3, that we can locate the boundary of Hopf instability (where the uniform state is unstable to a uniform perturbation and at which spatially uniform time-dependent oscillations set in). We can use the equations derived in 10.3.2 to draw similar loci for instability to spatial pattern formation. For this, we can choose a value for the ratio of the diffusivities / and then find the conditions where eqn (10.48), regarded as a quadratic in either y or n, has two real positive solutions. The latter requires that... 

Fig. 27. Spatial pattern formation within a 1.5 x 1-mm2 area of a Pt( 100) surface during kinetic oscillations as recorded by scanning photoemission microscopy. (From Ref. 138.)... 

It may appear that Table 1 contains an essentially complete summary of patterns that may form in electrochemical systems. This impression is misleading, since Table 1 only roughly summarizes results observed so far or predicted with models. These are investigations concentrating on phenomena that can be described with two essential variables (two-component systems). This survey is certainly not yet completed. Furthermore, numerous examples of current or potential oscillations involve complex time series. Only in a few cases does the complex time series result from the spatial patterns. In most cases, the additional degree of freedom will be from a third dependent variable, such as from a concentration that adds an additional feedback loop into the system, as discussed in Section 3.1.3. Spatial pattern formation in three-variable systems is an area that currently develops strongly in nonlinear dynamics. In the electrochemical context, the problem of pattern formation in three-variable systems has not yet been approached. 

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