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Study of the dynamic behavior of Three-Variable Autocatalator
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| Department: | Chemical Engineering |
| Organization: | National Institute of Applied Sciences and Technology |
| Department: | Chemical Engineering Department |
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2007-09-29
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The three-variable autocatalator is a prototype of complex dynamical behavior. Indeed, period doubling and chaos are found when the bifurcation parameter, mu, 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.
In the first notebook (showalter.nb), we try the following values of mu: 0.1, 0.14, 0.15, 0.151, and 0.155 to observe period 1, 2, 4, 8, and 5 behaviors, respectively. For mu=0.153, chaos is obtained and the phase-space graph is that of a strange attractor. When mu 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 notebook bifurcation3.nb, we show how to obtain the bifurcation diagram (Figure 3 page 5246 of Peng et al. (1990)).
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.
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bifurcation parameter, bifurcation diagram, chaos, period doubling, strange attractor, power spectrum, complex dynamic behavior, chemical oscillator, autocatalytic reaction, three-variable autocatalator
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| bifurcation3.nb (1002.5 KB) - Mathematica Notebook [for Mathematica 6.0] | | showalter.nb (2.5 MB) - Mathematica Notebook [for Mathematica 6.0] |
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