The following nonisothermal reaction system is theoretical. The steps are as follows:
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 the hypothetical chemical species , , and .
The autocatalytic reaction is the following step: , with catalyzing its own formation. This step introduces the nonlinear term in the governing equations. The last reaction, B → C + Heat is exothermic. The rate constant of the first reaction, P → A, follows the Arrhenius rate-law. Thus depends on the temperature.
The governing equations for the two intermediate species and the temperature are usually written in the form:
, , .
The dimensionless governing equations are:
, , . Here , , and are dimensionless concentrations of , , and the dimensionless temperature, and the four parameters , , , and depend on the rate constants of the individual reactions , , , and , the concentration of the precursor , the molar density , the molar heat capacity , the surface heat transfer coefficient , the surface area , the surrounding temperature , the heat of reaction for the reaction , and the activation energy of the reaction .
The Demonstration illustrates the dynamics of the concentrations , , and the temperature for various values of the bifurcation parameter . Choose "time series" to get a plot of versus time or "phase space" to get a three-dimensional parametric plot of .
For = 0.5586, the phase-space graph is that of a spiral attractor.
Reference: S. K. Scott and A. S. Tomlin, "Period Doubling and Other Complex Bifurcations in Non-isothermal Chemical Systems," Philosophical Transactions of the Royal Society of London, Series A: Mathematical and Physical Sciences, 332(1624), 1990 pp. 51–68.