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/ |