What does the weather, quantum theory, the financial market, biology, chemistry, capsizing ships, snooker, cauliflower�s, the universe and Jurassic Park all have in common? (apart from the fact that they where all just mentioned together in the previous sentence). They are all, in one way or another, linked to the subject of chaos.

But what is this strange phenomenon which everyone seems to know of but no one can explain. Take for example the aforementioned Jurassic Park. You may or may not remember that Jeff Goldblum played a Chaos mathematician in the film (the technical term for someone who studies chaos is actually a Chaologist but who cares). If we just put aside for one moment that no one could possibly make a living out of being a professional chaos mathematician, it would seem that chaos could just be a buzz word that American writers put into their books to make them sound more technical, but we can really explain what it is.

One example of chaos is the path that a leaf might take when floating downstream or the motion of a ball in a Roulette wheel. Predicting this motion is difficult at the best of times even in a simple situation, so could you ever predict the motion of the leaf or is it entirely random?

There are three things that make up the basis of chaos theory:

1: Chaos is deterministic, this means that the situation has something determining its behaviour
2: The situation in question is very very very sensitive to initial conditions
3: On the surface chaos appears to be random. However it is not random since it is deterministic.

Even some very simple situations show examples of chaotic behaviour. A pendulum may be considered to be the paragon of the regular Newtonian system. However if you introduce a second force on the pendulum, at some frequencies we gain a chaotic response.

For example if we have a simple pendulum swinging from left to right as we look at it, and a motor attached to the top of the pendulum moving in and out, at certain frequencies of the motor the pendulum will swing wildly and move in and out of various elliptical orbits. In this case small changes in the initial position of the pendulum give extremely varied results in terms of the swing of the pendulum.

So how can we describe or account for this strange behaviour, what makes the pendulum behave in this erratic way?
It all goes back to the three points that make up chaos. The pendulum is very sensitive to initial conditions, and small changes lead to large differences in its motion. It also appears to be moving in a random fashion. The most important thing is that it is deterministic. After all there is only two motions occurring, one left to right, the other forwards and back wards, and yet out of this simple situation we can find chaos.
Let us look at an even simpler situation so that we might better understand how this phenomenon can occur.
One of the specific situations where chaos occurs is in systems that rely on feedback (and example of this is the internal workings of an amplifier).
In mathematical terms a system which uses feedback is an iterative process. Take for example the equation:

Here the next term for X is found by a function "f" of the previous X. One now famous use of this simple method is in what is called the Logistic Map shown below:

This can be used for a starting value of t=0 to show how we arrive at chaotic situations. If we take X₀ to be 0.4 we will then vary the values for l . What we find is this, if l =2 the values of X will settle down to 0.5 exactly. If the value of l = 1+� 5, the values of X oscillate between 0.5 and 0.8, never settling down. If we increase l to equal 4 we get complete chaos. The value of X never settles down and we get what looks like completely random numbers (although they are not random since they have been determined by an equation).

This transition from calm to chaos, illustrates how even a simple equation can generate chaotic results. We can see chaos even more when we change the first X value from 0.4 to 0.35, a difference of only 0.05.

This small change in starting value means little difference when l is 2 and 1+� 5 however in terms of when l =4 the results deviate dramatically from those seen before and by about the 15^th iteration the two results are completely different. This demonstrates how sensitive a chaotic situation is to its initial conditions.

Another point about the logistic map is that as we increase l from the value of 2 we notice that a strange phenomenon called Feigenbaum doubling occurs. We saw that when l = 1+� 5 the values of X oscillated between two points. It is therefore said that this value has a period of 2 (a value of 4 for l has a period of 1). This carries on for further powers of 2, for example l =3.5 has a period of 4. The period of values of l continue to double in what is, surprisingly enough, called period doubling. This occurs with more frequency as the values of l tend towards two. This example of Feigenbaum doubling is a good model for similar situations in the human heart and the dripping of a tap where chaotic oscillations can also occur.

Imagine if you will a snooker table (the snooker table in the SA is a good example of a snooker table, so we'll use that). If you where feeling particularly sad, or wanted something to do for Maths coursework you might like to attempt to model the motion of the ball on that very table. How many variables would you have to consider to create a suitably accurate model? Believe it or not you would have to actually consider at least eight separate variables. 2 variables for the co-ordinates to mark the balls position on the table, 2 for the speed variables of the ball. If we also consider the spin on the ball as we should, we need a further 4 variables for the direction of the spin axis and the speeds related to spin. This is incidentally what makes snooker so difficult to play well (except when you miraculously set up a perfect potting opportunity for the opposition against all odds (sadly Sod�s law is not covered by chaos)). Anyway, now that we have our eight variables what do we do with them now? Well if we are looking at the motion of the ball over a period of time the best thing to do is plot a graph. But how do you plot an eight dimensional graph? You can�t (obviously) but you can plot certain values on different graphs (or you could take a Poincar� section, but you don�t even want to know what one of those is). The important thing is that there are only certain values that the variables can take, the ball can only have co-ordinates on the table, and there are only certain speeds that the ball can reach. If you try and imagine eight dimensional space, then there is only a certain zone which contains all the possible states of the snooker ball, ever. This is known in maths terms as the phase space of the snooker ball, and the path that a particular ball takes within phase space is called the phase portrait.

Moving away now from the snooker ball and looking in more general terms, the phase portrait is only one situation that the snooker ball can take. If we where looking at another system where there is a number of start positions we could plot them all on the same graph. What you might find depending on the situation is that all the possible points in the phase space eventually converge to a specific area where they continue to circulate. Such an area is called an attractor. A simple predator/prey system is a good example of this.
If the particular system you are studying is a steady state system (i.e. All the possible starting points eventually converge to one point) then the attractor of this system is a point in phase space.
If the system is periodic, oscillates between two ore more values (as with Feigenbaum doubling) the attractor in phase space is a loop or more commonly a torus (donut-see Mmm... Doughnuts, Issue 12). In fact all normal attractors of systems are tori, a point is a 0 dimensional torus.

So what about the chaotic motion we observed in the logistic map. Well an attractor for chaotic motion cannot be classified in the normal way. We thus call it a strange attractor. This is not because what the system is modelling is particularly strange but that the attractor in phase space is very strange. Computers are the most useful tool in producing pictures of attractors and are responsible for most of the developments of chaos over its history. Incidently a change in the nature of an attractor from say an ordered loop to a strange attractor is called bifurication (just in case you wanted to know).
Now for an example of a strange attractor.
It isn�t entirely certain where and to whom the origin of chaos can be attributed, but one man is known to have come up with the first attractor of its type. His name was Edward Lorenz and he was a weatherman in the 1960�s. Lorenz was interested in long term prediction of weather and so went about trying to model it with, amongst others, Newton�s laws. What Lorenz found was that when he came back to his computer one day, he wanted to begin his computations again. So he entered his previous value and started the program. What he saw was that the program diverged wildly from previous results. What he found was that the start value he had entered had been accurate to only three decimal places, where as the computer was working to six. The quick diversion from just a small difference in initial conditions was a sure-fire sign of chaos. Lorenz simplified his equations making them in terms of three variables (x, y and z). The result when plotted was what is now known as the Lorenz attractor (a mask/butterfly shaped form with no volume and an infinitely complex edge). The Lorenz attractor is an example of a strange attractor.

As well as modelling the weather, the equations in the Lorenz attractor also model a situation called the Lorenz water wheel. In this buckets are attached to a wheel (with me so far?). Water flows in at the top into the buckets turning the wheel round. As well as this the buckets have holes in the bottom which means that water drains out. If the flow of water at the top is right, the wheel will spin in a chaotic fashion rotating one way, slowing and then spinning the other, in accordance with Lorenz�s equations. It also makes a particularly nice water feature in a garden.

As a weather model the Lorenz attractor is not amazingly good but as an example it is useful. For example if we treat each side of the "mask" as different weather states (in terms of Britain this would probably be rain, or rain). A point on one side will either continue to circle one weather state or circle onto the other side. There is no telling when this might occur which is what makes long term weather forecasting difficult (and possibly why Michael Fish always seams to get it wrong).

It also means that you can never make an accurate prediction unless you know the precise initial conditions. It also means therefore that a slight change in the initial conditions could have large repercussions on the weather system. Hence the theory that if a butterfly flaps its wings in the Amazon, it may influence a tornado in Texas (i.e. it is influencing the initial conditions of the weather system). This also means that you to can influence the weather yourself. Try it right now, go outside and wave your arms about, you most certainly won�t look like you've gone completely mad I assure you. Go now!
Anyway now that you have come back inside, all be it a little out of breath, I shall continue.
So what other situations apart from the weather exhibit chaotic tendencies? As you might expect, there are any number of situations based on deterministic laws that are influenced by initial conditions, hence (or otherwise) there is a lot of chaos about.

There is even chaos in the universe itself. One of the things that people assume about the universe, and more closer to home, the solar system is that it is the typical Newtonian system, with the orbits of the planets fixed in stone, in summary, a clockwork universe. This is however far from the truth. Take as an example the solar system. Take as a further example Newton�s law of gravitation discussed in the article Weighing the Earth. If we apply this equation to even a simple system of three planets we find that the two body calculations are very complex. Since each planet is influenced by every other in the solar system, and to a larger extent the universe, calculations for just one planet are very complex. What we do find is that because of this many planets are in resonance with each other, for example Neptune has a 3:2 resonance with Pluto.

The Earth itself displays some chaotic principals also, this does not mean that we might be about to crash into Mars or spin off into the stellar void, but it does mean that an error as small as 15m in measuring the Earth's present position could make it impossible to predict its orbit in the long term future. This is because the Earth's orbit is slowly moving around 0.3⁰ per century, due to the motions of Jupiter.
The most interesting evidence of chaos in our solar system is in one of Saturn's moons, Hyperion. This potato shaped satellite has a regular orbit with Saturn but a chaotic spin due to its odd shape, which makes it look like it is "falling" through space.
At the other end of the scale we also find chaos closer to home, in the financial market. Any one who has seen activity at a stock exchange may have considered that to be obvious, but even in simple models of this principal, we find chaos. For example if we take a commodity like gold and try to model it, where "t" is the time in weeks and "p" is the price of gold, we might derive an equation like the one below:

Where A = price raised by the dealers

B = Number of people willing to pay the price of the dealers

This is an iterative equation like those seen before and models the price of the gold. Dealers always want to raise the price, and buyers are not willing to pay extortionate prices. In this equation if A-B >1 the price of gold will keep on increasing where as if A-B <1 there is a depression in the gold market, (probably coinciding with the death of Mr T or Ron Atkinson).
A better model for this would be one where the price the buyers where willing to pay was linked to the price set by the dealers (i.e. In terms of A). In this case we again get the now famous logistic map and chaos reigns again:

So even in the gold market, if the situation is right, we will find chaos. Add in all the other commodities and not just the precious metals and you have one complicated system.
Another simpler example is one that is used to demonstrate economic dynamics in some universities. It is called the beer distribution game. It is not usually played with actual beer involved, but I�ll let you make up your own extra rules at your discretion.
The situation is this. There are four stages in getting the beer to the consumer. The Brewer brews the beer. The Distributor distributes the beer to the Wholesaler who in turn delivers it to the Retailer, who sells it on to the Consumer. So the four players are Brewer, Distributor, Wholesaler and Retailer. Each player must try to have enough beer to satisfy demand without over or under-stocking beer, and so losing money. The game itself is played on a week to week basis.
In the usual scenario the game begins with a regular demand of four cases of beer from the retailer. After a few weeks the order of the consumer is changed from four cases to eight and remains at this value for the rest of the game. You would expect that each player would smoothly adjust their order to take into account the four extra crates. Instead huge oscillations occur in order value, and it has been known that by the 30^th week the distributor could be ordering as many as 40 crates even though there is still only a demand of eight on the retailer. Once again we see a simple deterministic system with feedback go wildly and unpredictably out of control. Starting to see a pattern yet?
We can also see this kind of oscillation in chemistry. A particularly interesting reaction is the Belousov � Zhabotinskii reaction (B-Z for short (thank god!)). Here is the recipe:

500ml sulphuric acid

14.30g malonic acid

5.22g potassium bromate

0.548g ammonium ceric nitrate

1-2ml ferroin

Serving suggestion: Strawberries

Hmm yummy! When mixed together in a beaker this reaction actually oscillates between red and blue with a period of about 1-minute. The colour change occurs due to the iron compound ferroin, which is magenta when in its reduced state and blue when oxidised (which is nice). When this reaction is done on a large scale, with reactants being constantly replenished we find that period doubling occurs as in the logistic map, and strange attractors are created.
To summarise so far from what we have looked at, it would seem that the situations most likely to give chaotic responses are those which rely on feedback or contain oscillations. Chaos can therefore also be found in many aspects of life, including fluid dynamics, amplifiers and circuits, particle accelerators, biology populations and engineering.
Let us change tack now and look at the slightly less useful but more beautiful side of the whole chaos topic.

Warning this section contains graphic use of the word "fractal" which some may find disturbing.

I have put this warning in since the word stated above is one used far to often in the subject of chaos and so I will attempt to limit its use to places where it is only extremely necessary. The word in question is actually derived from a Latin word, "Fractus", meaning broken up and irregular, and created by none other than Benoit Mandelbrot. Fractal geometry is the geometry of deterministic chaos since none of the complex systems here could hope to be modelled purely by Euclidean geometry of circles and triangles. It also just so happens that a number of the strange attractors of certain systems are fractals, for example the mask shaped Lorenz attractor discussed earlier.
Fractals also occur in nature, ferns, Cauliflower�s and Broccoli are natural fractals as well as clouds, mountains and lightning.
The difference between a fractal and a normal geometrical figure is that when you get close to the edge of a square, it looks like a straight line. A fractal has infinitely complex edges, so as you zoom in, the edge never becomes a simple straight line. As we shall see, there can even be repetition of the original structure on the edges of itself.
A good example of a simple fractal is the Sierpinski gasket. Take a normal equilateral triangle. Cut out the middle part of this triangle with another, whose corners touch the original triangle�s edge. You thus gain three little triangles. Perform the same operation again and you get nine smaller triangles. The continued repetition of this process produces a simple fractal, which as you zoom in looks like itself at every level. This is also a good example of infinity; the fact being that eventually, after infinitely many iterations there will be no area left at all.

Some fractals like the Sierpinski gasket are based on chaotic or non-linear systems, while others are based on random processes. An example of this is Diffusion Limited Aggregation (DLA). This works on a simple random principal. If you imagine a giant chessboard with one queen surrounded by four pawns. The queen cannot move but the four pawns move in a random, "drunkard�s walk". If they reach the solitary queen again, they turn into a queen themselves and cannot move. Another three pawns are also created and start moving. The pattern created from the DLA is much like the pattern seen when lightning discharges, or how water seeps through rock.

The most famous of all the fractals must be the Mandelbrot and Julia sets, named after the French mathematicians, Benoit Mandelbrot and Gaston Julia. They are both linked to each other but I will start with the Julia set. There are infinity many Julia sets, all based on one initial equation.

The rule goes that if Z_k does not go to infinity by this iteration, it is a member of the Julia set. Different sets are derived from different values of C. This looks pretty simple you may well be saying smugly to yourself (or out loud so everyone can hear, I just don�t know). However it is worth saying that the value of Z is not a simple number it is a complex number, of the nature: z = x+yj, where j is � -1. Thus both the Julia and Mandelbrot sets occupy the complex plane (for more information on complex numbers, either wait until the further maths book of the same name, or alternatively if you haven�t got a year to waste see the article on complex numbers in issue 12 or read a book).
Different values of C in the Julia set produce different Images, from a contained area to small islands. As before the edge of this set is infinitely detailed.

The other set is the Mandelbrot set. The Mandelbrot set is the set of all C values where the Julia set breaks up into islands to form a Cantor set, it is the boundary between order and chaos. It is based on the equation:

If C_k does not go to infinity, C₀ is a member of the Mandelbrot set. The Mandelbrot set itself requires 6,000,000 calculations which shows why computers are so useful in this kind of situation, and how sadistic you would have to be to carry out the operation by hand. The Mandelbrot set can easily be recognised by its "bug shape". The most amazing thing about this set is that as you zoom in on areas you see the same "bug shape" reoccurring within itself. For more detailed and probably more comprehensible information on these sets and an amazing calculator program so you to can draw fractals, (just take my word for it, it really is great) see the article in this very issue.

To see the Julia sets corresponding to different points on the mandelbrot set, click here.

Undoubtedly the most enduring thing about chaos is its beauty shown by fractals such as these, which make mathematics seem a magical subject once again.
So what have we learned from all of this. The universe isn�t as simple as people would have us believe (we knew that). But in the future will it really be possible to determine the weather months in advance? Assuming we had the technology to measure all the positions of every particle and calculate its movement would we be able to determine the cosmic future? The answer is no. The one thing that has put pay to the idea of a clockwork universe is quantum physics. Heisenberg�s uncertainty principal states that everything we can measure is subject to truly random fluctuations like the moment of decay of a radioactive nucleus. Thus there are a number of probable states, in which a particle can exist, which means it is never truly possible to know the outcome of a situation exactly, no matter how much detail you know, even down to the level of the motions of particles. It also means that in the end it all comes down to statistics (how unfortunate) and what probable energy levels a particle might have.
So if we haven�t learnt very much, what have we found out? That some situations can be modelled by simple equations, some display chaotic behaviour and still more are too complicated to be modelled successfully at all. There is still a lot of development still to be found in the theory of chaos, but it has meant that maths is not bounded by simple linear situations, and can attempt to predict the real world. It has also given rise to beautiful pictures, which have their base in both the arts and the sciences.
It just confirms what many people have been saying for centuries, its all just complete and utter chaos.