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# Bifurcation Diagram for the Three-Variable Autocatalator

This Demonstration shows how one can compute the bifurcation diagram for a nonlinear chemical system such as the three-variable autocatalator (see Details).
Indeed, in order to obtain such a diagram, one has to locate the maxima of the time series for all values of the bifurcation parameter, , which can be readily done using Mathematica's built-in function FindArgMax.
The Demonstration illustrates the dynamics of the concentrations , , and for various values of the bifurcation parameter . The time series option gives a plot of versus time and shows the loci of all maxima.
Try the following values of : 0.1, 0.14, 0.15, 0.151, and 0.155 to observe period 1, 2, 4, 8, and 5 behaviors, respectively. For , chaotic behavior occurs. When is large enough, you can observe a reversed sequence leading back to period 1 behavior. These results are confirmed by the bifurcation diagram (a remerging Feigenbaum tree), first given in [1] and reproduced in the present Demonstration.

### DETAILS

Consider a chemical reaction such as the shift conversion: . To study its chemical kinetics, it is written in abstract form as . The rate of this reaction is given by , where is the rate constant and , , and are the concentrations of the species , , and .
Hypothetically, the autocatalator's steps are the following:

Here is a precursor with constant concentration, is the final product, , , and are intermediate chemical species, , , , , , and are rate constants for the individual reactions, and , , , , and are the concentrations of the hypothetical chemical species , , , , and .
This hypothetical reaction scheme is a model for a three-variable autocatalator. The autocatalytic reaction occurs in the step , with catalyzing its own formation. This step introduces the nonlinear term in the governing equations, necessary in order to obtain sufficient complexity for chaos to occur.
The rate equations for the three intermediate species can be written in the form:
,
,
.
The corresponding dimensionless rate equations are:
,
,
,
where , , and are the dimensionless concentrations of , , and , and the four parameters , , , and depend on the rates of the individual reactions and the concentration of the precursor.
This is the bifurcation diagram:
Reference
[1] B. Peng, S. K. Scott, and K. Showalter, "Period Doubling and Chaos in a Three-Variable Autocatalator," The Journal of Physical Chemistry, 94(13), 1990 pp. 5243–5246.

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