### Highlights from Bifurcation diagram for the three-variable autocatalator

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# Bifurcation diagram for the three-variable autocatalator

14 Oct 2007 (Updated )

Computes the bifurcation diagram for the three-variable autocatalator

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Description

The three-variable autocatalator is a prototype of complex dynamical behavior. Indeed, period doubling and chaos are found when the bifurcation parameter, nu, is varied between 0.10 and 0.20.

The autocatalator's steps are the following:
P->A
P+C->A+C
A->B
A+2 B->3 B
B->C
C->D
Where P is a chemical precursor, D is a final product and A, B, C are intermediate species.

The autocatalytic reaction is the following step: A + 2 B -> 3 B with B catalyzing its own formation. This step introduces a nonlinear term in the governing equations that is necessary to obtain the complex dynamical behavior such as chaos.

One should try the following values of nu: 0.1, 0.14, 0.15, 0.151, and 0.155 to observe period 1, 2, 4, 8, and 5 behaviors; respectively. For nu=0.153, chaos is obtained and the phase-space graph is that of a strange attractor. When nu 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) given in Peng et al. (1990).

In the present code, it is shown how to obtain the bifurcation diagram (Figure 3 page 5246 of Peng et al. (1990)).

The code takes several hours (approximately 10 hours on a Pentium III 800 MHz) to compute the bifurcation diagram. No attempt to optimize the code was done.

Reference:
Bo Peng, Stephen K. Scott, and Kenneth Showalter, "Period Doubling and Chaos in a Three-Variable Autocatalator", The Journal of Physical Chemistry. Vol. 94, No. 13, 1990.

Here is a link to a Mathematica 6.0 Demonstration concerning the three-variable autocatalator:

http://demonstrations.wolfram.com/AStudyOfTheDynamicBehaviorOfAThreeVariableAutocatalator/

Required Products Optimization Toolbox
MATLAB release MATLAB 7.0.1 (R14SP1)
13 Sep 2008

It's unfortunate that the code has no comments to explain what it is doing. It's well over a year today since i wrote to the author asking for clarifications on some aspects of the code. Till date he/she has not replied.

10 Jun 2008

Not only the code is not commented and unclear, but what it claims to reproduce is not the exact picture of the referenced paper.

15 Oct 2007

corrected typo in description

15 Oct 2007

corrected typo in description

16 Oct 2007