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Homoclinic behavior in the Belousov-Zhabontinski reaction

Christophe LETELLIER
03/03/2008
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Boris Belousov (1930-1998)

In 1950’s that Boris Belousov (1930-1998) [1] observed the first self-sustained chemical reaction. It was based on a somewhat complicated chemical reaction: the oxidation of organic molecules by bromate ions BrO3-, oxidized by the redox pair Ce3+/Ce4+. This reaction can be reduced to [2]

3 CH2(COOH)2 + 2 HBr + HBrO3 = 3 CHBr(COOH)2 + 3 H20 .
It was subsequently studied by Anatol Zhabotinski (1938-) who
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Anatol Zhabotinsky

confirmed the oscillating nature of this chemical reaction in 1964 [3]. This reaction is now called the Belousov-Zhabotinski reaction. The network of chemical reactions is somewhat complicated and involves a large number of intermediate reaction species.

It was not until 1975 that the first observation of chaotic behavior in the Belousov-Zhabotinski reaction took place. This was done by John Hudson’s group. Since there was no understanding of chaotic behavior at that time, these results were not published. In fact, all the work carried out in this field was on periodic oscillations. This changed only only after the publication of Rössler’s work on oscillating chemical reactions [4]. This work used sets of ordinary differential equations to describe hypothetical chemical reactions and proposed the use of computers for numerically integrating these equations to show chaotic behavior. Rössler also mentioned that the Belousov-Zhabotinski reaction was a good candidate for finding chaotic behavior because of the observations by Arthur Winfree [5] that irregular oscillations were seen in this reaction. On reexamining their old data Hudson and his colleagues had the satisfaction of discovering plots like those shown in Fig. 1 that showed ``a time dependence, on average stable, but not periodic’’ [6].

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Fig. 1 Chaotic time series from the BZ-reaction
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Françoise Argoul

Following this, a multitude of behaviors typical of chaotic systems were observed: intermittency, period-doubling. One of these is very typical: it was studied by Françoise Argoul [7]: the trajectory slowly winds away from the fixed point at the center of the attractor, only to be reinjected back to the neighborhood of the focus and begin a new cycle (Fig. 2). This is called a homoclinic orbit: Poincaré had already discovered these at the heart of the three body problem. This type of chaos is somewhat difficult to analyze but it is important because it is one of the situations where a mathematical proof that

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Homoclinic chaos measured by F. Argoul

chaos exists can be carried out: the first was that of Poincaré. A normal form model was proposed to described the oscillations in the concentration of Ce4+ by Argoul, Arnéodo and Richetti. A topological analysis and a global model were investigated from this data set [8]. A three branch template was obtained and a set of three ordinary differential equations were obtained.

[1] B. P. Belousov, A Periodic Reaction and its mechanism, Sbornik Referatov po Radiatsionni Meditsine, Medgiz, Moscou, 1958, p. 145.

[2] A. H. Zaikin & A. M. Zhabotinskii, Concentration Wave Propagation in two-dimensional liquid-phase self-oscillating system, Nature, 225, 535-537, 1970.

[3] M. A. Zhabotinski, Periodic Processes of the oxydation of malonic acid in solution, Biofizika, 9, 306-311, 1964.

[4] O. E. Rössler, Chaotic Behavior in simple reaction system, Zeitschrift für Naturforsch A, 31 (3-4), 259-264, 1976 - O. E. Rössler, Chemical Turbulence: chaos in a simple reaction-diffusion system, Zeitschrift für Naturforsch A, 31 (10), 1168-1172, 1976.

[5] A. Winfree, Scroll-Shaped Waves of chemical activity in three dimensions, Science, 181, 937-939, 1973.

[6] R. A. Schmitz, K. R. Graziani & J. L. Hudson, Experimental Evidence of chaotic states in the Belousov-Zhabotinskii reaction, Journal of Chemical Physics, 67 (7), 3040-3044, 1977.

[7] F. Argoul, A. Arnéodo & P. Richetti, Experimental Evidence for homoclinic chaos in the Belousov-Zhabotinskii reaction, Physics Letters A, 120 (6), 269-275, 1987.

[8] C. Letellier, J. Maquet, H. Labro, L. Le Sceller, G. Gouesbet, F. Argoul & A. Arnéodo, Analyzing chaotic behaviour in a Belousov-Zhabotinskii reaction by using a global vector field reconstruction, Journal of Physical Chemistry A, 102, 10265-10273, 1998.

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