Cooperative Dynamics of Coupled and Forced Oscillators
Following spatiotemporal and spatial patterns on single electrodes, described in Chapter 2, the cooperative dynamical behavior of coupled electrochemical dynamical systems is analyzed. Simple introductory theoretical analysis of the coupling is followed b
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Cooperative Dynamics of Coupled and Forced Oscillators
3.1 3.1.1
Coupled Oscillators Outline Theoretical Aspects of Coupling the Electrochemical Oscillators
The formation of dissipative patterns on electrode surfaces is related to various kinds of spatial couplings that interact with the electrochemical process in a way dependent, among others, on the geometrical arrangement of the electrodes and the operational mode of the electrochemical experiments. A natural development of the concept of individual reaction sites engaged into such couplings is the case of coupled oscillators. It is thus not surprising that the treatment of coupling of the individual oscillators and spatiotemporal phenomena in a single oscillator may have common points. The analysis of such complex systems allows to understand better the general mechanisms underlying the spatiotemporal phenomena, including the coupling that exist between oscillators in living systems, e.g., in the biological cell membranes. The coupling of chemical or electrochemical oscillators can lead to either more complex or simpler oscillatory regimes, in the latter case including even ceasing of the oscillations. A special variant of such coupling can be realized through the external perturbation of a single oscillator or of a set of them. Karantonis and Nakabayashi [1] have analyzed the case of coupling of two limit cycle electrochemical oscillators, described by only two dynamical variables. In this work the diffusive coupling in one dynamical variable is considered which, under appropriate conditions, can lead to dephasing of the oscillators. This particular problem is related to the suggestion by Kuramoto [2] that chemical spatiotemporal chaos might arise where diffusive coupling leads to dephasing. The problem is important since it was shown, in terms of the model approach, that the diffusive coupling of the transmembrane voltage, in the presence of a strong deformation of the phase flow in the vicinity of the limit cycle, can cause the dephasing of oscillators and bursting oscillations [3]. The model of electrochemical M. Orlik, Self-Organization in Electrochemical Systems II, Monographs in Electrochemistry, DOI 10.1007/978-3-642-27627-9_3, # Springer-Verlag Berlin Heidelberg 2012
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3 Cooperative Dynamics of Coupled and Forced Oscillators
oscillator, described by Karantonis and Nakabayashi, is simplified: only one electroactive species is present which is produced on the electrode surface and also carries all current in the solution via diffusion and migration. The linear approximation of the concentration distribution in the diffusion layer (Nernst concept) is also applied. The electrochemical process is characterized with the N-NDR region, and the total current consists of the faradaic and the capacitive currents. Accordingly, as described in Chap. 2 of volume I, the dynamics of such a system is ruled by the two ordinary differential equations of a general form, in terms of dimensionless variables (note that for consistence with the figures, the orig
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