# Bifurcation Diagram for the Three-Variable Autocatalator

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.

μ

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.

α

β

γ

μ

logγ

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.

μ

μ=0.153

μ