Catastrophe and propagation in chemical reactions

Abstract

Multiple time scales, common in known chemical instability systems, lead to quasi-steady state ’’behavior’’ surfaces in the phase space of chemical concentrations. A result of catastrophe theory is used to classify the types of topological features, the ’’elementary catastrophes,’’ that these surfaces may take on. By using a multiple time scale scheme we predict the wave form and velocity of a variety of propagating phenomena that can occur because of the various types of catastrophes such as the cuspoids and umbilics. Since the state of chemical equilibrium is unique, we show that the presence of catastrophes on the behavior surface is strictly due to nonequilibrium processes on a short time scale. An important implication of this work is that the topological analysis of multiple time scale systems unfolds the richness of the potential for the variety of propagating phenomena in reacting diffusing systems.

Dynamics of spatially nonuniform patterning in the model of blood coagulation

“…The relationship between the topological properties of slow manifolds of the chemical system ͑catastrophes͒ and the type of wave or pulse propagation has been considered in studies 30,31 by Ortoleva. Unlike his models, our model is constructed without the use of slow manifold formalism in the explicit form.…”

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“…La clásica teoría del fenómeno de precipitación espacial periódica explicada por Ostwald (1925) y Prager (1956 ha sido completada en las últimas décadas con el estudio del comportamiento de las ondas químicas l (Prigogine y Nicolis, 1967;Flicker y Ross, 1974;Ortoleva y Ross, 1972, 1973, 1975Schmidt y Orto1eva, 1977;Feinn y Ortoleva, 1977;Feinn et a1., 1978;Lovett et a1., 1978;Schmidt y Ortoleva, 1979, 1980, 1981Feeney et a1., 1983;Ortoleva y Schmidt, 1985) en el que se han introducido los conceptos de reacciones autocatalíticas, mecanismos de autoalimentación (<<feedback,,), crecimiento competitivo de partículas y ciclos de sobresaturación-nucleación-deplección (ciclo de Ostwald-Liesegang).…”

Winfree Meandering in a 2-Dimensional 2-Variable Excitable Medium

“…A multiple time scales approach (cf. [10]) would be desirable, but is so far applicable only outside the core region. This is because the wave has to be assumed to propagate into a region with invariant excitability properties (so that the velocity of propagation becomes a constant calculable as an eigenvalue of the scaled equation [10]).…”

“…[10]) would be desirable, but is so far applicable only outside the core region. This is because the wave has to be assumed to propagate into a region with invariant excitability properties (so that the velocity of propagation becomes a constant calculable as an eigenvalue of the scaled equation [10]). Winfree [19,12] observed that the trajectories in local phase space are all closed in the core region of a non-stilf system, forming a smooth concentric set when the behavior at all local points is entered into the same picture: Figure 8.…”

“…Heart tissue, with the pace-maker system removed, provides a classical example (see van der Pol and van der Mark [3], Wiener and Rosenblueth [4], Swain and Yanagita [5], Balakovsky [6], and Krinsky [7]). A quantitative description of trigger waves as "moving catastrophes" [8,9] was recently given by Feinn and Ortoleva [10].…”