Cruising a Strange Attractor

A Spreadsheet Simulation of Chemical Chaos


Introduction

The Belousov-Zhabotinsky (BZ) reaction is probably the most famous chemical reaction exhibiting chaotic behaviour. Various simplified models of the 11 variable mechanism have been presented, which reproduce oscillations and chaotic behaviour of the reaction.

This project was aimed at producing a simple and accessible representation of the chaotic behaviour using spreadsheets. One-Dimensional or Return Maps, plotting a variable against its next value in time, are an efficient means of representing chaos. As will be demonstrated here, these maps are linear for first and second order reactions, oscillate for oscillatory behaviour and show a distinct turning point for chaotic behaviour.

Numerical integration, using the Simple Euler Method, was applied to first and second order reactions as well as a simplified, two-dimensional Oregonator model and a three-dimensional model of the BZ reaction as suggested by Field and Gyoergi (1992).


First Order Reaction

The decomposition of Azomethane has been chosen as an example of a first order reaction. A plot of concentration versus time is linear and yields a linear return map.

Second Order Reaction

The decomposition of NO3 is used as an example of a second order reaction. Again, the plot of concentration versus time is linear and yields a linear return map.

Simplified, Two-Dimensional Oregonator

Finding y, which corresponds to the concentration of Br-, from numerical integration, the plot of y vs. z displays nice oscillations and so does the return map.

Three-Dimensional Model of the BZ reaction

Field and Gyoergi (1992) placed their main emphasis on investigating different flow rates in a CSTR-reactor. Here two different flow rates have been chosen to exhibit chaotic behaviour, simply titled high and low flow rate. All three variables present in this model have been numerically integrated, keeping one variable constant for all calculations, increasing one by small increments (cf. Simple Euler Method) and hence working out the third one. As can be seen by clicking on the relevant links, many of the two variable plots show sudden jumps (remember, we are in three-dimensional space and slicing through a plane of the variable held constant) and the return maps display distinct turning points.

- high flow rate, numerical integration of v, plot of v vs. z, return map, v

- high flow rate, numerical integration of x, plot of x vs. z, return map, x

- high flow rate, numerical integration of z, plot of z vs. x, return map, z

- low flow rate, numerical integration of v, plot of v vs. z, return map, v

- low flow rate, numerical integration of x, plot of x vs. z, return map, x

- low flow rate, numerical integration of z, plot of z vs. x, return map, z

If you would now like to have a closer look at the sheets themselves, try varying initial values and constants in the numerical integrations:

-click here if your computer runs msworks for windows.

-click here if your computer runs msexcel.

-click here if your computer runs lotus 1-2-3.

-click here if your computer runs msworks for dos.

-click here if your computer runs works 3.0 for macintosh.


Some References

Chemical Chaos:

S. Scott, "Clocks and chaos in chemistry", New Scientist, 2 December 1989, pp. 53-59

I. R. Epstein, "Chemical Chaos", Chemistry & Industry, 4 March 1991, pp. 157-162

State-of-the-Art Symposium: "Self-Organization in Chemistry", Journal of Chemical Education, Vol. 66, No. 3, March 1989

J. L. Hudson, J. C. Mankin, "Chaos in the Belousov-Zhabotinkii reaction", J. Chem. Phys. 74(11), 1 June 1981, pp. 6171-6177

Oregonator:

J. Rinzel, W. C. Troy, "Bursting phenomena in a simplified Oregonator flow system model", J. Chem. Phys. 76(4), pp. 1775-1789

R. J. Field, R. M. Noyes, "Oscillations in chemicals systems. IV. Limit cycle behaviour in a model of a real chemical reaction", J. Chem. Phys., Vol. 60, No. 5, 1 March 1974, pp. 1877-1884

Three-Dimensional Model of the BZ Reaction:

L. Gyoergi, R. J. Field, "A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction", Letters to Nature, Vol. 355, 27 February 1992, pp. 808-810

L. Gyoergi, R. J. Field, "Simple Models of Deterministic Chaos in Belousov-Zhabotinsky Reaction", J. Phys. Chem. 1991, 95, pp. 6594-6602

Related Sites

Polymerizations Coupled to Oscillating Reactions

Chaos in Chemical Systems

Chemical Oscillations and Waves in the Physical Chemistry Lab

Computer Simulation and Study of the Belousov-Zhabotinsky Reaction

B. P. Belousov: The Way to the Discovery