Maxima and Minima for the Three-Variable Autocatalator
Peng et al.  derived three coupled nonlinear ODEs to describe the behavior of the three-variable autocatalator. This Demonstration plots the loci of the maxima and minima of the time series . Such information is used to map the next maximum or minimum point. For particular values of the bifurcation parameter (for example, ), the result is chemical chaos, and a map of the next maximum/minimum shows the typical behavior of a chaotic attractor. On the other hand, when periodic behavior is observed (e.g., for ), a map of the next maximum/minimum displays only a small number of isolated points.
Consider a chemical reaction such as the exchange reaction: . 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:
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:
, , .
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.
 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. doi:10.1021/j100376a014.