Homoclinic behavior in the Belousov-Zhabontinski reaction

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]

It was subsequently studied by Anatol Zhabotinski (1938-2008) who

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].

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

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. A 3D normal form for this type of data is introduced here.