Theoretical Background of Spatial and Spatiotemporal Patterns in Dynamical Systems

Principles of formation of spatiotemporal and spatial patterns in chemical and electrochemical systems are outlined. Types of chemical waves: phase, kinematic, and trigger ones are distinguished. The formation of spatiotemporal patterns in excitable chemi

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Theoretical Background of Spatial and Spatiotemporal Patterns in Dynamical Systems

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Chemical Reaction–Diffusion Systems Basic Characteristics of Spatiotemporal Instabilities

In volume I we described the dynamical systems, the entire state of which was the same at a given time: the whole system either remained in the steady state or was oscillating with the same period and amplitude and the oscillation shape, irrespective of the location. Such situation corresponds to perfect mixing of homogeneous solutions or, in the case of heterogeneous processes, to perfectly uniform catalyst or electrode surfaces. However, when the solution is not perfectly mixed, or is not mixed at all, or the solid catalyst or electrode surface of realistic structure is in contact with reacting solution, local differences (gradients) of concentrations of species and/or of the electric field are no longer homogenized and give rise to the diffusion and/or migration transport components directed parallel to such surfaces. In this way different sites of the system can “communicate” with each other, exchanging information about their present states, i.e., there appear certain couplings between them, giving rise to various dissipative patterns. At a first glance, this conclusion may appear surprising, as the diffusion is usually considered a process that homogenizes a system, and does not assist in keeping or, all the more, developing any inhomogeneity. However, under nonequilibrium conditions, the structure-forming role of diffusion (and other kinds of transport), in cooperation with the chemical or electrochemical process of appropriate characteristics, becomes evident, both experimentally and theoretically. Another type of patterns can emerge if the spatial symmetry of the system is broken by the convective transport which self-organizes into various patterns; the pattern of convective flows determines then the spatial distribution of the system components. Due to their specific characteristics, patterns of convective origin will be described in a separate Chap. 5.

M. Orlik, Self-Organization in Electrochemical Systems II, Monographs in Electrochemistry, DOI 10.1007/978-3-642-27627-9_1, # Springer-Verlag Berlin Heidelberg 2012

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1 Theoretical Background of Spatial and Spatiotemporal Patterns

One should emphasize that the mechanisms and conditions for the formation of spatial and spatiotemporal patterns, the latter ones meaning the concentration fronts traveling through the system and called also “chemical waves,” are of tremendous importance for understanding of such phenomena in chemical and biological systems. For example, the biological morphogenesis, when initially identical stem cells differentiate into the specialized tissues may be imagined as the effect of the onset and the progress of the traveling wave of chemical species (like Ca2+ ions) which carries information on further different development of initially identical cells. The onset and development of concentric blue wave fronts of the oxidized (Fe3+) form of the

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