Crystal spiral pattern study

These principles also apply to spatial distributions of populations as reported by Hassell et al. (1991). In a study using host-parasite interactions as the model, a variety of spatial patterns were developed using the Nicholson-Bailey model. Host-parasite interactions demonstrated patterns ranging from static "crystal lattice" patterns, spiral waves, chaotic variation, or extinction with the appropriate variation of only three parameters within the same set of equations. The deterministically determined patterns could be extremely complex and not distinguishable from stochastic environmental changes. 

Begishev et al. (1985) studied anionic polymerization fronts with e-caprolactam. There were two interesting aspects to this system. First, the polymer crystallizes as it cools, which releases heat. Thus, a front of crystallization follows behind the main front. Volpert et al. (1986) investigated this two-wave system. Second, a hot spot moved around the front as it propagated down the tube, leaving a spiral pattern in the product. The entire front propagated with a velocity on the order of 0.5 cm min which was a function of the concentrations of activator and catalyst. The hot spot circulated around the outside of the 6-cm (i.d.) front sixteen times as rapidly as the front propagated. 

In this respect, feedback-mediated parametric modulation seems to be a more promising control strategy, since in this case the modulation period always coincides exactly with the actual rotation period of the spiral wave [20, 21]. Another important motivation to study feedback-mediated dynamics of spiral waves is related to the fact that a feedback is either naturally present or can be easily implemented in many excitable media [22-25]. For example, recent experimental investigations performed with the BZ medium [26, 27] and during the catalytic CO oxidation on platinum single crystal surfaces [28] reveal that global feedback can provide an efficient tool for the control of pattern formation. 

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