# Bifurcation Diagram for the Three-Variable Autocatalator

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This Demonstration shows how one can compute the bifurcation diagram for a nonlinear chemical system such as the three-variable autocatalator (see Details).

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Contributed by: Housam Binous, Zakia Nasri, and Brian G. Higgins (April 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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.

## Permanent Citation