**Jeff HughesRoyal Melbourne Institute of Technology**

The basis of modern Chaos theory is the search for underlying order in apparent randomness. The common idea of chaos is that of a system without any apparent pattern or order - out of control. This seems to imply that a system behaving this way has no underlying rules governing its behaviour. There is also a common belief that complicated behaviour is associated with complicated systems of controlling rules, while simple sets of rules lead to simple behaviour.

The modern theory of Chaos has shown both these ideas to be flawed. Model systems based on a few basic rules can show extremely complex behaviour. This behaviour may have no obvious pattern yet is not random since the system is driven by a set of rules and is hence predictable (Note 1) (but only if the initial conditions can **precisely** be specified).

Chaos is thus more correctly referred to as **Deterministic Chaos** (an apparent contradiction in terms!). Chaotic behaviour of this type is referred to as **aperiodic**, in contrast to random. The problem is then for the experimenter to decide whether a dynamic system, behaving without apparent order or pattern, is behaving randomly (unpredictably) or chaotically (there are underlying rules governing the motions).

The meteorologist, Edward Lorenz, showed in 1963 that this optimism is ill-founded (Note 2). Lorenz simplified his model to the stage it only contained three variables:

dx/dt = a(x-y) ...................(1)dy/dt = r*x - y - x*z ..........(2)

dz/dt = b*z + x*y .............(3)

It need not concern us at this stage what x, y and z refer to. In fact, they are composite variables, combining the real variables of the system such as temperature, pressure etc. How x, y and z change with time will be reflected in changes in the real system variables.

This is a system of **differential** equations - they reflect how x, y and z will change with time. It is a very simple set of equations, but the set has two important characteristics:

- the system is non-linear (due to the terms xz and xy appearing in equations 2 and 3)
- the system has no exact (analytic) solution - i.e. we cannot write down a set of equations in x, y and z which, differentiated, give equations 1 - 3.

How can we then study the behaviour of such a system? Fortunately there are approximate (or numeric) methods which can help us. With the help of a computer - a PC is quite adequate (Note 3)- the way x, y and z change with time can be studied. such plots are called time series (see figure 1a). Typical results for x, y and z show that they undergo oscillations in a fairly regular fashion, until the onset of chaotic or aperiodic oscillations.

The modern PC is, however, far in advance of the primitive computer available to Lorenz in 1963. Even the printing out of numbers was a slow process so, to save time, he only printed their the first 4 decimal places. This round-off error turned out to be fortunate, as it led to an accidental discovery of the crucial feature of this system.

Lorenz's computer did not have a graphics screen - just a line printer, to print out lists of numbers. One day he decided to check the results of a run, so he re-entered a set of values of the variables from part-way through a run. At first, the numbers produced were the same as his first run, but gradually started to differ until the two sets of numbers became quite different (see figure 1b). Why? Lorenz realised that the values of x,y and z he re-entered were not exactly the same as the first run, but only correct to 4 decimal places. Because of this rounding-off error the true value of x was slightly different. Thus the key element of chaotic systems is demonstrated - sensitivity to initial conditions.

Can a Cockroach Crawling up a Wall in Melbourne Cause a Cyclone in Cairns?**climate**. At this stage we also come back to the question asked earlier - how can the experimenter, presented with data which has no simple repeating pattern, distinguish chaotic behaviour from truly random behaviour?

The answer is to change the way we look at the data. Observing how the variables change with time as in figure 1, called a **time series**, is not the best way to detect underlying order. Supposing instead of time series we plot a series of points (x,y,z) at successive times. The plot is called a **phase portrait**. For the Lorenz model a characteristic shapes emerges. In figure 2 the projection of this 3-D image onto a 2-D surface is shown. The shape basically consists of two lobes (the Lorenz 'butterfly'), but the line never crosses itself. The shape is arrived at regardless of initial values of (x,y,z). The unpredictability comes from the fact that two adjacent tracks can at some future stage diverge onto different lobes. These tracks, or 'orbits', can be infinitesimally close, but never cross. Consider then two points on nearby tracks (corresponding to two initial sets of values, differing by very small amounts). At first the paths followed by each point will be closely aligned but at some future time one track can branch off into the other lobe (Note 4).

These shapes traced out by the (x,y,z) values in phase space are called **attractors** (so named because the system seems drawn or attracted to this region of phase space). If the attractor is not a point or a closed loop it is **strange**. For truly random data there will not be any distinguishing pattern in phase space. Thus, the existence of a strange attractor is a pointer to the existence of some underlying order to the system. Despite apparent erratic behaviour the motion is 'attracted' towards a regulating order by a set of rules. The experimenter can then look for a model, or set of mathematical equations, to study the system. In our example of weather models, the climate may be thought of as a weather attractor. If such an attractor does exist, then it is most likely to be strange.

Is There Any Strangeness in Chemistry?

The reason that the Lorenz model was chosen to introduce chaos is that the 3 equations used in the model have direct relevance to chemistry. If we let x, y and z represent concentrations of chemical species in a reaction then the study of the rate of change of these variables is what we are studying in **Chemical Kinetics**. The 3 Lorenz equations could be sets of equations describing a chemical mechanism, and the dx/dt, dy/dt and dz/dt are the rates of reaction with respect to each of these variables. The vast majority of chemical reactions do not behave like the Lorenz model (fortunately!). They are not sensitive to initial conditions - imagine how difficult life would be if changing the amount of reactant by a microgram changed the likely course of a reaction! The 'attractor' for chemical reactions is a single point - the equilibrium point (see figure 3a).

Is it possible, then, for a chemical system to have a different kind of attractor, or even behave chaotically? Theoretical models have been proposed which have non-linear elements similar to the Lorenz equations (Note 5) and when the equations for these systems are integrated they show **oscillatory** behaviour (see figure 3b). The attractor for an oscillatory or periodic system is a closed loop.

Examples of chemical reactions which can oscillate have been known from 1828. A reaction in an electrochemical cell which produced an oscillating current was described by Fechner. Several other examples of oscillations in heterogeneous systems were reported during the nineteenth century, but the first report of oscillations in a closed homogeneous system was reported by Bray in 1920, the same time that the theoretical Lotka mechanism was reported (Note 5). Not much notice was taken of these reported reactions because they appeared to violate the Second Law of Thermodynamics. However what the Second Law really forbids is oscillations around the point of equilibrium - like the damped oscillations of a spring. A chemical system can oscillate providing the system is **far from equilibrium**. It is now recognised that there are three requirements for chemical oscillations to occur:

- the reaction must be far from equilibrium.
- the reaction must have autocatalytic steps. An autocatalytic reaction contains steps of the type: A + X → 2X so the formation of X as a product catalyses its reaction. Thus the system has a chemical form of feedback.
- the system must be able to exist in two steady states (neither of which is the true equilibrium state).

_{3}

^{-}) and a ceric salt. The 1951 paper was rejected, even though it contained a recipe for the reaction, because the editor of the journal thought it was impossible! Another Russian, Zhabotinskii, took up the work and replaced citric acid by malonic acid. He published several papers on what is now known as the Belousov-Zhabotinskii (BZ) reaction. An attractive variation on their original recipe is to add the redox indicator, Ferroin. This produces oscillations in colour from blue-green to red (the 'Oscillating Traffic Light' reaction) (Note 7).

Oscillations in such systems as the BZ and Bray reactions can be studied in a number of ways. Oscillations in colour can be followed by monitoring the absorbance of the solution in a spectrophotometer. Oscillations in chemical concentrations can be followed by monitoring the chemical potential of the solution, using an inert electrode (Pt, graphite) and a reference electrode. In either case, regular oscillations are observed.

Does Chaos Develop From Order?To avoid this depletion the reactions were studied in a **Continuous Stirred Tank Reactor (CSTR)**. Reagents are pumped in through three or four lines using a single peristaltic pump. The vessel has a fixed volume, and excess mixture is allowed to overflow. Rather than try and monitor the effect of changing initial concentrations of the several reactants the system is studied for sensitivity to only one parameter - the **average residence time** the solution is in the cell. This can be controlled by the pumping rate. Such systems are then quite sensitive to residence times and can exhibit periodic or aperiodic behaviour, depending on the pumping rate.

In the Lorenz system, evidence of chaotic behaviour was seen in the existence of a strange attractor in, phase space. The problem with the BZ reaction study is that generally the change of only one variable with time is studied (eg. absorbance, cell potential etc.). To obtain another variable for plotting in phase space the value of the variable is measured after a fixed time delay. This gives a series of points (X_{t}, X_{t + T}) where T is the fixed time delay. The choice of T is somewhat arbitrary, but there does appear to be an optimum delay of about one tenth to half the average period of oscillation. When data from aperiodic regions of the BZ reaction in a CSTR are plotted in this way, a strange attractor similar to figure 3(c) is observed. Thus the oscillations observed can truly be described as chaotic and not just random.

The question still remains as to the origins of this chaos. Does it arise from the chemical reactions themselves or is it a product of some inhomogeneity in the system such as non-uniform mixing, temperature fluctuations etc? The full mechanism proposed for the BZ reaction by Field, Koros and Noyes (FKN mechanism) contains 18 steps and 21 independent species. A much simpler model, called the Oregonator (Note 8), can be constructed from the FKN model by combining variables and making certain assumptions. This model also contains the nonlinear elements and autocatalytic steps required for oscillating reactions and Chaos.

Although the Oregonator model appears quite simple, mathematical treatment of it is not. If integration of the equations by simple methods such as the Euler method (see Note 1) is attempted the variables do oscillate but at rapidly increasing amplitude until the 'out of bounds' message is seen on the computer. This is clearly not realistic results from the huge difference in the magnitude of the rate constants (varying from 10^{-10} to 10^{9} in some versions of the model). Such equations are known as stiff equations and require special techniques to integrate. Merely taking smaller time intervals does not get round the problem.

Using these special techniques initial reports did describe aperiodic behaviour and the existence of strange attractors. Later reports have claimed that the aperiodicity is a numerical artefact, and by modifying the techniques used only strict periodic behaviour is observed.

An alternative approach to model the observed chaotic behaviour was taken by Rossler. Rather than try and model the chemical reactions taking place he set up a series of equations motivated by the fluid dynamics governing the chemical reactions in a CSTR (Note 9). These equations are much simpler to integrate, and exhibit oscillatory behaviour and a strange attractor similar to the attractor observed for the experimental data. The position can thus be summarised as that Chaos has undoubtedly been observed experimentally but the jury is still out on whether the 'chemical' or 'physical' interpretation of this Chaos is the correct one.

The BZ reaction, and similar reactions are by far the most-studied examples of Chaos in Chemistry but there are other examples (Note 10). Some gas-phase reactions are autocatalytic and have the potential for chaotic behaviour. A simple example is the reaction between hydrogen and oxygen in a CSTR. Other examples involve heterogeneous systems, such as the catalytic oxidation of carbon monoxide on platinum and the dissolving of metals such as copper and nickel in solution when an electric potential is applied. Examples in Biochemistry include the interconversion of adenosine triphosphate (ATP) and adenosine diphosphate (ADP) via glycolysis. In view of the way that enzymes are recycled efficiently in biochemical reactions it seems likely that many more examples of oscillatory mechanisms in Biochemistry will appear. An example of chaotic behaviour in this context is the occurrence of Chaos in heartbeats, or fibrillation. The question arises as to whether other examples of a 'breakdown' in a biological function might be due to a transition from periodicity to chaos.

**Notes**

- An example of a complex system governed by a single, simple rule is the logistic equation devised by the biologist R.M. May:X
_{n+1}= r*X_{n}*(1 - X_{n})This is a crude model of population growth, incorporating a 'feedback' factor. X represents a fractional population, and the (1 - X) term reflects a limiting factor such as food supply. Experimenting, even with a calculator, can show the complexity of this system. With X_{)}= 0.4 and r = 2, the system quickly becomes stable (settles on a fixed value of 0.5). For r = 3, the system oscillates between two values (period 2). For r = 3.5 the system repeats after every 4 values (period 4). For r = 3.6 there is no discernible pattern (Chaos). Also for the chaotic regions (eg. r = 3.6) the system exhibits extreme sensitivity to initial conditions. Changing X_{0}to 0.4000001, the numbers produced initially are very close but gradually diverge till after about 50 steps they are quite different.Behaviour of the system as r changes is displayed in the famous 'bifurcation' diagram (see James Gleick's book "Chaos", or the program FRACTINT. These are described in the references). - A good historical account of this discovery is given in James Gleick's book "Chaos".
- The simplest method for integrating differential equations is Euler's method:Suppose y changes with time, then dy/dt ~ (y
_{2}-y_{1})/(t_{2}-t_{1})so y_{2}~ y_{1}+ (t_{2}-t_{1})*dy/dtApplying this method to Lorenz's first equation, in a computable form:y_{new}= y + dt*a*(x-y) where dt is the fixed time interval. Using a computer, dt can be made very small for greater accuracy. - The property that tracks in phase space, initially close to each other, can diverge exponentially with time is mathematically described by the Lyupanov exponent. If d
_{0}is the distance between two points initially, then at a later time:dt = d_{0}*2*t if >0 then the system is chaotic - The LOTKA model is as follows:A + X → 2X , dx/dt = k1ax - k2xyX + Y → 2Y , dy/dt = k2xy - k3yY → P , da/dt = -k1ax , dp/dt = k3y
- The following is a recipe which produces a good visual demonstration of the Bray reaction, or 'Oscillating Iodine Clock'.Prepare the following solutions:(1) 25mL of 30% H
_{2}O_{2}+ 75mL of H_{2}O(2) 2.1g of KIO_{3}+ 80mL of 2M H_{2}SO_{4}(3) 8.0g malonic acid + 0.2g MnSO_{4}in 100 mL H_{2}Ocombine solutions 1-3, plus a few mL of 3% starch solution, in a 500mL measuring cylinder. As the reaction oscillates blue-black bands of the I_{2}starch complex travel up the column. - The following recipe for the 'Traffic Light' reaction is one I have found convenient for classroom demonstrations. It avoids the need to prepare solutions beforehand and the dilution of the sulfuric acid provides heat for the reaction. The only disadvantage of this method is the need to handle concentrated sulfuric acid.To 250mL of water in a beaker, add 25mL of concentrated sulfuric acid (take care!). With rapid stirring, add to the hot solution 1g of ceric sulfate, 1.5g of potassium bromate and 3g of malonic acid. Oscillations from dark to pale yellow occur after a few seconds. If ferroin indicator is added (a few mL) the 'traffic light' oscillations from blue-green to violet are produced.
- The OREGONATOR mechanism is as follows: A + Y → XX + Y → PB + X → 2X + Z2X ⌢ A + Pz → fY(A,B,P and X represent various forms of Bromate, Y is Br
^{-}, Z is Ce^{4+}and f is a factor representing the organic species.) - The ROSSLER equations are: dx/dt = -y - zdy/dt = x + 0.398*ydz/dt = 2+ z*x -4*z
- A good discussion of the examples can be found in Scott's book "Chemical Chaos".

**Software**

- James Gleick, Chaos: The Software.This is a superb set of programs, produced by the author of the best-selling text on chaos. The package includes 6 programs: Mandelbrot sets, Magnets and Pendulums, Strange Attractors, The Chaos Game, Fractal Forgeries and Toy Universes. Requires EGA/VGA monitor.
- Roger Stevens, Fractal Programming in PascalText + disk of Pascal programs. The text covers the maths of the fractals, programming techniques and source code. The programs are written for EGA/VGA screens but can easily be adapted for CGA. Of particular interest are the fractal landscape programs.
- Fractint.This is a sensational program, the more so for the fact that it is free! The authors state "don't send money. We only want recognition". A large variety of fractals and strange attractors are included, and as the program uses integer arithmetic it is much faster than other programs (without needing a coprocessor chip). Images (B&W) can also be printed to an Epson or HP laser printer. The images look superb in VGA but all screen types (even B&W CGA) are catered for. The latest version (v. 15) is 380K when decompressed, so a high density drive or hard disk is required. However, an earlier version (v.10) fit comfortably on a 360K disk.

**Texts**

- Gleick, J., "Chaos", Sphere Books, 1988.

An excellent summary of the topic, with a good historical background. This paperback actually made the best-seller list. - Stewart, I., "Does God Play Dice?" The Mathematics of Chaos, Blackwall, 1989.

This is also a very readable survey of Chaos. Stewart has a more eccentric approach to the topic than Gleick, but the two texts together provide a good basic library of chaos. - Moon, F., "Chaotic Vibrations", Wiley, 1987.

Gives a good, though fairly technical, coverage of the analysis of chaotic vibrations in physical and chemical systems. It provides a good insight into some of the experimental and computational techniques used to explore Chaos. - Avnir, D., (Ed), "The Fractal Approach to Heterogeneous Chemistry", Wiley, 1989

The text explores the applications of fractal geometry to areas of chemistry such as the surfaces of electrodes and colloids, polymer growth, proteins, chromatography and geochemistry. - Scott, S, K., "Chemical Chaos", Oxford, 1991.

An excellent, though technical, coverage of applications of chaos theory to chemical systems. - Hao Bai-Lin, "Chaos II", World Scientific, 1990.

A collection of most of the key research papers in the Chaos field.

**Articles**

*Chaos:*

- Crutchfield, J.P., Farr, J.D., Packard, N.H., Shaw, R.S., "Chaos", Sci. Amer., 255,38 (1986)
- Rossler, O.E., Wagman, K., "Chaos in the Zhabotinski Reaction", Nature, 271,89 (1978)
- Argoul, F., Arnedo, A., Richetti, P., Roux, J.C., "Chemical Chaos: From Hints to Confirmation", Acc. Chem. Res., 20, 436 (1987)
- Epstein, L R., "Oscillations and Chaos in Chemical Systems", Physica 7D, 47 (1987)
- Epstein, L R., "Patterns in Time and Space", Chem. & Eng. News, 24 (Mar. 30 1987)
- Scott, S., "Clocks and Chaos in Chemistry", New Scientist, 31 (2 Dec 1989)
- Mandelbrot, B., "Fractals - A Geometry of Nature", New scientist, 29 (15 Sept.1990)
- May, R.M., "Simple Mathematical Models With Very Complicated Dynamics", Nature, 261, 459 (1976)
- Stewart, I., "Portraits of Chaos", New Scientist, 22 (4 Nov 1989)
- Glasser, L., "Order, Chaos and All That!", J. Chem. Ed., 66, 997 (1989)

*Oscillating Chemical Reactions:*

- Bunting, R.K., "Periodicity in Chemical Systems", Chemistry 45, 18 (1972)
- "Oscillating Reactions", J. Chem. Ed.,65, 1004 (1988)
- Regan, H., "Oscillating Chemical Reactions in Homogeneous phase", J. Chem. Ed., 49, 302 (1972)
- Field, R.J., "A Reaction periodic in Time and Space", J. Chem. Ed., 49, 308 (1972)
- Lefelhocz, J.F., "The Colour Blind Traffic Light", J. Chem. Ed., 49, 312 (1972)
- Walker, J., "Chemical Systems That Oscillate", Chem. Tech., 320 (May, 1980)
- Aroca, P., Aroca R., "Chemical Oscillations: A Microcomputer-controlled Experiment", J. Chem. Ed.,64, 1017 (1997)
- Cartwright, H.M., Farley, H. A., "Why Do some Reactions Oscillate?", Education in Chemistry, 46 (March, 1989)

About the Author

Jeff Hughes obtained his PhD in inorganic chemistry from Latrobe University in 1975. Following a period of research work at Melbourne University, he has taught chemistry at RMIT in Melbourne and Capricornia Institute in Rockhampton. Currently he lectures in physical and environmental chemistry at RMIT. His current research interests are in the applications of numerical methods of studies of problems in solution chemistry and studies of the behaviour of aluminium in treated water.