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Self-Organization in Electrochemical Systems II pp 1–63Cite as

Theoretical Background of Spatial and Spatiotemporal Patterns in Dynamical Systems

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Part of the book series: Monographs in Electrochemistry ((MOEC))

Abstract

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 chemical media, in which local coupling of different sites of the reaction medium occurs through diffusion, is described. General principles of linear stability analysis of the spatially extended systems are presented, including conditions for the formation of Turing patterns. Extension of such analysis for electrochemical systems includes the role of (1) nonlocal coupling, realized through migration and of (2) positive and negative global couplings, dependent on the galvanostatic or potentiostatic operational mode, and the spatial arrangement of the working, reference, and auxiliary electrodes. All possible combinations of interactions of these couplings with the processes characterized with the negative differential resistance of N-NDR and S-NDR type were analyzed as a source of respective patterns. In particular, for the S-NDR systems the conditions for the formation of Turing patterns were outlined.

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Notes

  1. 1.

    for the Dirichlet boundary conditions, defined as zero perturbations at z = 0 and z = L borders, \( u_{\rm n} =\sin(n{\pi} z/L);\) n = 1,2,3, …

  2. 2.

    Codimension of bifurcation means the number of system’s control parameters which have to be tuned to achieve the bifurcation. The Hopf and saddle-node bifurcations are codimension-1 bifurcations. Besides the Turing-Hopf bifurcation, another codimension-2 bifurcation occurs at a cusp point in which two lines of saddle-node bifurcations meet tangentially.

  3. 3.

    Although usually Turing patterns are considered to be stationary, it is also possible to consider nonstationary Turing patterns (see further in this section).

  4. 4.

    strictly speaking, electrode potential E = φ dl–φ dl(RE), but since φ dl(RE) = const, the dynamics of dE/dt = dφ dl/dt

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  1. Faculty of Chemistry, University of Warsaw, Warsaw, Poland

    Marek Orlik

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Orlik, M. (2012). Theoretical Background of Spatial and Spatiotemporal Patterns in Dynamical Systems. In: Self-Organization in Electrochemical Systems II. Monographs in Electrochemistry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27627-9_1

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