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.