# A Study of the Dynamic Behavior of a Three-Variable Autocatalator

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

### DETAILS

A chemical reaction such as the shift conversion: can be written in chemical kinetics as . The rate of this single chemical reaction is , where is the rate constant and , , and are the concentrations of the chemical species , , and .
The following reaction system is theoretical. The autocatalator's steps are the following:

Here is a chemical precursor with constant concentration, is the final product, , , and are intermediate chemical species, , , , , and are rate constants for the reactions, and , , , , and are the concentrations of hypothetical chemical species , , , , and .
This hypothetical reaction scheme is a model for a three-variable autocatalator. The autocatalytic reaction is the following step: , with catalyzing its own formation. This step introduces the nonlinear term in the governing equations that is necessary in order to obtain complex dynamical behavior such as chaos.
The rate equations for the three intermediate species are usually written in the form:

The dimensionless rate equations are:

Here , , and are the dimensionless concentrations of A, B, and C, and the four parameters , , , and depend on the rates of the individual reactions and the concentration of the precursor.
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, while the phase-space option gives a three-dimensional parametric plot of .
You should 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 = 0.153, chaos is obtained and the phase-space graph is that of a strange attractor. 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) given in 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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