Some information on chaos

Mathematical chaos has been a hot topic in recent years. If you have seen Jurassic Park, you probably know a little about it already. Chaos is a property of some deterministic systems. A deterministic system is one in which future states depend strictly on the current conditions. They can be modelled by dynamical systems. Historically the idea has been that all processes ocurring in the universe are deterministic, and that if we knew enough of the rules governing the behavior of the universe and had measurements about its current state we could predict what would happen in the future. These ideas have been applied with a great deal of success to falling objects, tide prediction, and many other systems. However, there have always been things which we have never been able to predict, things such as the weather. This was written off for a long time as being due to our incomplete knowledge of the system. With the development of chaos however, a new idea has emerged.

The essence of chaos is something called sensitive dependence on initial conditions. For the systems in which our ability to make predictions has been good, only a reasonable approximation of the inital state is neccessary for prediction. The paths, or orbits, of inital conditions which are nearby to one another stay close together. A reasonable approximation of the current state yeilds a reasonable approximation of the future state. With sensitive dependence, this is not the case. In a system which exhibits sensitive dependence on initial conditions, reasonable approximations of the initial state do not provide reasonable approximations of the future state. The orbits of nearby initial conditions diverge until one can no longer discern any indication that they were once similar to one another. In order to make useful long term predictions in such a system, one needs measurements of initial conditions with infinite accuracy, which are impossible to obtain. This means that systems with a deterministic underpinning can generate behavior which seems random and unpredictable. A good example of such a system is the logistic map

The idea of deterministic chaos did not come to full fruition until after the advent of the computer, which could be used to model systems which were previously unapproachable by traditional methods. One of the most common methods of modelling a system is by way of a phase space diagram. A phase space diagram is a diagram created by plotting the various dependent variables of a system against one another. For example, one could create a phase space diagram by plotting a moving object's position versus its velocity in two dimensions, or even by plotting its position vs. velocity vs. acceleration in three dimensions. The end result is something that looks like a single point moving along a path through space (this is the orbit that I spoke of earlier). This orbit can do several things: it can settle down to one point and stop, it can travel around in a circle, it can exhibit behavior which is predictable but not quite repetitive. It can also behave chaotically by travelling unpredictably on a path within a certain region of space, known as a chaotic attractor. Chaotic attractors can take may forms, but they all fall into a category of objects known as fractals.

Chaos theory is fairly broad, abstract topic. I'm not expert on it, but we did (attempt to) learn a bit about it in math class.

What I basically got out of it, is that chaos theory deals with systems so complex, involving so many variables and interactions between them, that they literally appear to any human observer to have entirely random final outcomes. It also deals with simple systems that invariably end up complexifying themselves, as varibles introduce other variables (note the butterfly example above). Chaos theory is often illustrated with fractals, geometric constructions that begin as simple elements, but wind up having so many variables that each individual progression seems to be random, yet in the end they do form a clear, repeating pattern. You could apply chaos theory to many real world concepts, including weather patterns, astronomical phenomena, the human body, random number generators, and so on.

As for Jurassic Park, Ian Malcolm supposedly was into chaos theory (at least in the book, he was). Michael Critchon likened the events on the island to a chaotic system, having so many variables, yet a clear formula for success or failure. As you can probably see, this has fairly wide-ranging implications, as we deal with chaotic systems daily. In fact, you could say that our entire lives are really insanely complex chaotic systems. Our lives start out simple, and spontaneously complexify. One variable introduces another, which introduces two more, and so on.

You could even say the same about the universe itself.

"Chaos theory" was a term invented by the popular press to describe the part of dynamics usually referred to as Chaos. To quote Ian Stewart, "Chaos no more deserves to be isolated as a theory in its own right than "skeleton theory" deserves to be isolated from zoology."

Nonlinear dynamics is the "newest" self-contained theory in this area, as it has recently taken off in its own right. When people say chaos theory, they often mean nonlinear dynamics.

As for a definition of Chaos, the following was proposed at a meeting of the Royal Society in 1986. It is hard to categorise such a new field, but this was the best attempt -

(Math) Stochastic behaviour occuring in a deterministic system.

This appears to be a paradox (stochastic means random), and can be paraphrased as "Lawless behaviour governed entirely by law." It is a common misconception that only complex systems demonstrate Chaos, when in fact something as simple as kx2 - 1 does.

Chaos Theory

To those who professionally study the Chaos theory, it is a fundemental way of perceiving the world around them. However, for the rest of us it seems to be a mystery, one that boggles minds while simotaniously stimulating them to great extents. For someone to understand the theory could prove quite revolutionary, as the chaos theory is not obvious at first, rather it is somewhat abstract. Understanding this emmensly complex notion will vastly improve a persons ability to fully comprehend- and appriciate - the universe, which is after all, the main goal of science.

An extremely vital concept of this theory is the “butterfly effect.” A man by the name of Edward Lorenz discovered this on his weather simulating computer. Keep in mind that this was in the 1970’s and the power of computers at the time was relatively minute compared to their power today... On this computer he could graph the weather patterns (in his computer generated universe) months in advance. The computer read out for the patterns went to four decimal places, a fairly accurate and precise measurement for the time, however when the program was actually operating it would use many more. One day he decided to check the accuracy of his program. Now, Lorenz had great faith in his creation but, as anyone with an anaylitical mind, he wanted to prove to himself just how close to perfection his computer was. He decided to collect a set of data representing all of the current conditions of his “world” and ran the program. Then, when he attempted a repeat of the weather cycle he was very surprised at the results. At first, the graphs of both patterns were very closely related but as more and more time went by, they gradually became more and more different, thus causing the graphs to become more and more dissimilar. Remember how the computer only remembered conditions to four decimal places? This was the cause of the chaos that he experienced. Although the initial conditions in the repeated experiment were exact to one ten-thousandth of one, as the amount of time increased so did the difference in the graphs. This effect came to be known as the butterfly effect. To apply the butterfly effect to a hypothetical situation: Say that something as small as a thunderstorm occurs . In the very distant future, perhaps even thousands of years, the world wide weather will be severely different than it would be had the storm never transpired (remember the word of advice: if you go back in time, don’t touch anything). Even in ancient folklore, the butterfly effect is portrayed:

For want of a nail, the shoe was lost;

For want of a shoe, the horse was lost;

For want of a horse, the rider was lost;

For want of a rider, the battle was lost;

For want of a battle, the kingdom was lost;

After deriving this butterfly effect, Edward Lorenz decided to look for chaos in other places than simply his computer world. This lead him to convection. He took a liquid in a container and place it on a stove, then he turned the heat on. Very early in the experiment the liquid began to boil (this was expected), and by boiling the liquid began to swirl in one direction (this was also expected). As the heat intensified and the liquid approached higher and higher temperatures the boiling liquid went into a frenzy with extreme palpatations and the swirling became faster. A few minutes later as the heat reached even higher levels he noticed that the swirling began to slow down. This surprised him, but not as much as what was to come. Eventually the liquid stopped swirling, this only lasted for a few moments however as the most chaotic action quickly occurred. The liquid began to spin in the opposite direction. The graph of that liquid convection became rather famous in the world of chaos and is known as the Lorenz Attractor. This graph appears to have intersecting lines, however they do not intercept each other due to the fact that it is a three- dimensional graph and the seemingly intersecting lines are actually at different altitudes. This graph was made by using the following variables: rate of spin and the height of the palpatations. This experiment was the first of many to follow, in a way, it was the chaos theory summed up in a graph.

Next, it is important to understand the fractal. These are infinitely iterated geometric series, or infinitely self repeating structures in a finite area. They are yet another way of representing chaos without words. When you see a fractal, at first it may appear to be very confusing and completely random. However, upon further anyilization you will notice that there is organization to it. And, as with everything dealing with chaos, fractals can be seen quite often in nature. For example, in trees, fractals exist in many places. Next time you see a tree, closely examine its branches, notice that each individual branch seems to closely assimilates the one preceding it thus forming one giant fractal. The root system of a tree also follows this same pattern. Even in the leaves of a tree, fractals are frequently found. Another example in a slightly different way would be a rock. As you zoom more and more closely to the surface of a rock, you will eventually notice a very distinctive arrangement of the surface. This you may notice is infinitely self repeating, one of the main aspects of a fractal.

This theory of chaos, an oxymoron in and of itself is a very complicated and confusing idea. However, when broken down into easier terms to understand, anything, including this theory can be addequately understood.

Chaos Theory: A Brief Introduction


What exactly is chaos? Put simply, it is the idea that it is possible to get apparently random results from normal equations. The techniques of chaos theory also cover the reverse: finding the order in what appears to be random data.

The Butterfly Effect: Sensitive Dependence on Initial Conditions

When was chaos first discovered? The first true experimenter in chaos was a meteorologist, named Edward Lorenz. In 1960, he was working on the problem of weather prediction. He had a computer set up, with a set of twelve equations to model the weather. It didn't predict the weather itself, but it did predict the weather of a hypothetical meteorological system.

One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the number off his printout and left to let it run.

When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly out of sync from the original. Eventually he figured out what happened. The computer stored the numbers to six decimal places in its memory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typed the first three digits, .506.

By all conventional ideas of the time, it should have worked. He should have gotten a sequence very close to the original sequence. A scientist considers himself lucky if he can get measurements with accuracy to three decimal places. Surely the fourth and fifth, impossible to measure using reasonable methods, can't have a huge effect on the outcome of the experiment. Lorenz proved this idea wrong.

This effect came to be known as the butterfly effect. The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings.

The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)
This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment. Such things are impossible to avoid in even the most isolated lab. With a starting number of 2, the final result in a chaotic system can be entirely different from the same system with a starting value of 2.000001. In many experiments, it is simply impossible to achieve this level of accuracy - just try and measure something to the nearest millionth of an second!

From this idea, Lorenz stated that it is impossible to predict the weather accurately. However, this discovery led Lorenz on to other aspects of what eventually came to be known as chaos theory.

The Lorenz Attractor

Lorenz started to look for a simpler system that had sensitive dependence on initial conditions. His first discovery had twelve equations, and he wanted a much more simple version that still had this attribute. He took the equations for convection, and stripped them down, making them unrealistically simple. The system no longer had anything to do with convection, but it did have sensitive dependence on its initial conditions, and there were only three equations this time. Later, it was discovered that his equations precisely described a water wheel.

At the top, water drips steadily into containers hanging on the wheel's rim. Each container drips steadily from a small hole. If the stream of water is slow, the top containers never fill fast enough to overcome friction, but if the stream is faster, the weight starts to turn the wheel. The rotation might become continuous. Or if the stream is so fast that the heavy containers swing all the way around the bottom and up the other side, the wheel might then slow, stop, and reverse its rotation, turning first one way and then the other. (James Gleick, Chaos - Making a New Science, pg. 29)

The equations for this system also seemed to give rise to entirely random behavior. However, when he graphed it, a surprising thing happened. The output always stayed on a curve, a double spiral. There were only two kinds of order previously known in mathematical systems: a steady state, in which the variables never change, and periodic behavior, in which the system goes into a loop, repeating itself indefinitely. Lorenz's equations were definitely ordered - they always followed a spiral. They never settled down to a single point, but since they never repeated the same thing, they weren't periodic either. He called the image he got when he graphed the equations the Lorenz attractor.

In 1963, Lorenz published a paper describing what he had discovered. He included the unpredictability of the weather, and discussed the types of equations that caused this type of behavior. Unfortunately, the only journal he was able to publish in was a meteorological journal, because he was a meteorologist, not a mathematician or a physicist. As a result, Lorenz's discoveries weren't acknowledged until years later, when they were rediscovered by others. Lorenz had discovered something revolutionary; now he had to wait for someone to discover him.

Another system in which sensitive dependence on initial conditions is evident is the flip of a coin. In one model of this system, there are two variables in a flipping coin: how soon it hits the ground, and how fast it is flipping. Theoretically, it should be possible to control these variables entirely and control how the coin will end up. In practice, it is impossible to control exactly how fast the coin flips and how high it flips. It is possible to put the variables into a certain range, but it is impossible to control it enough to know the final results of the coin toss.

Fractals

Biological Systems

A similar problem occurs in ecology, and the prediction of biological populations. The equation would be simple if population just rises indefinitely, but the effect of predators and a limited food supply make this equation incorrect. A simple model that takes this into account is the following:

next year's population = r * this year's population * (1 - this year's population)

In this equation, the population is a number between 0 and 1, where 1 represents the maximum possible population and 0 represents extinction. R is the growth rate. The question was, how does this parameter affect the equation? The obvious answer is that a high growth rate means that the population will settle down at a high population, while a low growth rate means that the population will settle down to a low number. This trend is true for some growth rates, but not for every one.

One biologist, Robert May, decided to see what would happen to the equation as the growth rate value changes. At low values of the growth rate, the population would settle down to a single number. For instance, if the growth rate value is 2.7, the population will settle down to .6292. As the growth rate increased, the final population would increase as well. Then, all hell broke loose. As soon as the growth rate passed 3, the line broke in two. Instead of settling down to a single size, the population would jump between two different sizes. It would be one value for one year, go to another value the next year, then repeat the cycle forever. Raising the growth rate a little more caused it to jump between four different values. As the parameter rose further, the line bifurcated, or doubled, again. The bifurcations came faster and faster until suddenly, chaos appeared. Past a certain growth rate, it becomes impossible to predict the behavior of the equation. However, upon closer inspection, of the graph of final solutions, it is possible to see white strips where the population never reaches certain sizes. Looking closer at these strips reveals little windows of order, where the equation goes through the bifurcations again before returning to chaos. This self-similarity, the fact that the graph has an exact copy of itself hidden deep inside, came to be an important aspect of chaos.

Market dynamics and Coastlines

An employee of IBM, Benoit Mandelbrot was a mathematician studying this self-similarity. One of the areas he was studying was cotton price fluctuations. No matter how the data on cotton prices was analyzed, the results did not fit the normal distribution. Mandelbrot eventually obtained all of the available data on cotton prices, dating back to 1900. When he analyzed the data with IBM's computers, he noticed an astonishing fact:

The numbers that produced aberrations from the point of view of normal distribution produced symmetry from the point of view of scaling. Each particular price change was random and unpredictable. But the sequence of changes was independent on scale: curves for daily price changes and monthly price changes matched perfectly. Incredibly, analyzed Mandelbrot's way, the degree of variation had remained constant over a tumultuous sixty-year period that saw two World Wars and a depression. (James Gleick, Chaos - Making a New Science, pg. 86)
Mandelbrot analyzed not only cotton prices, but many other phenomena as well. At one point, he was wondering about the length of a coastline. A map of a coastline will show many bays. However, measuring the length of a coastline off a map will miss minor bays that were too small to show on the map. Likewise, walking along the coastline misses microscopic bays in between grains of sand. No matter how much a coastline is magnified, there will be more bays visible if it is magnified more.

Fractional Dimensions

One mathematician, Helge von Koch, captured this idea in a mathematical construction called the Koch curve. To create a Koch curve, imagine an equilateral triangle. To the middle third of each side, add another equilateral triangle. Keep on adding new triangles to the middle part of each side, and the result is a Koch curve. A magnification of the Koch curve looks exactly the same as the original. It is another self-similar figure.

The Koch curve brings up an interesting paradox. Each time new triangles are added to the figure, the length of the line gets longer. However, the inner area of the Koch curve remains less than the area of a circle drawn around the original triangle. Essentially, it is a line of infinite length surrounding a finite area.

To get around this difficulty, mathematicians invented fractal dimensions. Fractal comes from the word fractional. The fractal dimension of the Koch curve is somewhere around 1.26. A fractional dimension is difficult to conceive, but it is possible to understand what's going on to some extent. The Koch curve is rougher than a smooth curve or line, which has one dimension. Since it is rougher and more crinkly, it is better at taking up space. However, it's not as good at filling up space as a square with two dimensions is, since it doesn't really have any area. So it makes sense that the dimension of the Koch curve is somewhere in between the two.

Fractal has come to mean any image that displays the attribute of self-similarity. The bifurcation diagram of the population equation is fractal. The Lorenz Attractor is fractal. The Koch curve is fractal.

During this time, scientists found it very difficult to get work published about chaos. Since they had not yet shown the relevance to real-world situations, most scientists did not think the results of experiments in chaos were important. As a result, even though chaos is a mathematical phenomenon, most of the research into chaos was done by people in other areas, such as meteorology and ecology. The field of chaos sprouted up as a sideline research area for scientists working on problems that were related to it.

Later, a scientist by the name of Feigenbaum was looking at the bifurcation diagram again. He was looking at how fast the bifurcations come. He discovered that they come at a constant rate. He calculated it as 4.669. In other words, he discovered the exact scale at which it was self-similar. Make the diagram 4.669 times smaller, and it looks like the next region of bifurcations. He decided to look at other equations to see if it was possible to determine a scaling factor for them as well. Much to his surprise, the scaling factor was exactly the same. Not only was this complicated equation displaying regularity, the regularity was exactly the same as a much simpler equation. He tried many other functions, and they all produced the same scaling factor, 4.669.

This was a revolutionary discovery. He had found that a whole class of mathematical functions behaved in the same, predictable way. This universality would help other scientists easily analyze chaotic equations. Universality gave scientists the first tools to analyze a chaotic system. Now they could use a simple equation to predict the outcome of a more complex equation.

Many scientists were exploring equations that created fractal equations. The most famous fractal image is also one of the most simple. It is known as the Mandelbrot set. The equation is simple: z=z2+c. To see if a point is part of the Mandelbrot set, just take a complex number z. Square it, then add the original number. Square the result, then add the original number. Repeat that ad infinitum, and if the number keeps on going up to infinity, it is not part of the Mandelbrot set. If it stays down below a certain level, it is part of the Mandelbrot set. The Mandelbrot set is the innermost section of the picture, and typically, different colors represent how quickly a point diverges from the set. One interesting feature of the Mandelbrot set is that the circular humps match up to the bifurcation graph of the population growth equation. The Mandelbrot fractal has the same self-similarity seen in the other equations. In fact, zooming in deep enough on a Mandelbrot fractal will eventually reveal an exact replica of the Mandelbrot set, perfect in every detail.

Conclusions

Fractal structures have been noticed in many real-world areas, as well as in mathematician's minds. Blood vessels branching out further and further, the branches of a tree, the internal structure of the lungs, graphs of stock market data, and many other real-world systems all have something in common: they are all self-similar.

Scientists at UC Santa Cruz found chaos in a dripping water faucet. By recording a dripping faucet and recording the periods of time, they discovered that at a certain flow velocity, the dripping no longer occurred at periodic intervals. When they graphed the data, they found that the dripping did indeed follow a pattern.

The human heart also has a chaotic pattern. The time between beats does not remain constant; it depends on how much activity a person is doing, among other things. Under certain conditions, the heartbeat can speed up. Under different conditions, the heart beats erratically. It might even be called a chaotic heartbeat. The analysis of a heartbeat can help medical researchers find ways to put an abnormal heartbeat back into a steady state, instead of uncontrolled chaos.

Other researchers discovered a simple set of three equations that graphed a fern. This started a new idea - perhaps DNA encodes not exactly where the leaves grow, but a formula that controls their distribution. DNA, even though it holds an amazing amount of data, could not hold all of the data necessary to determine where every cell of the human body goes. However, by using fractal formulas to control how the blood vessels branch out and the nerve fibers get created, DNA has more than enough information. It has even been speculated that the brain itself might be organized somehow according to the laws of chaos.

Chaos even has applications outside of scientific research. Computer art has become more realistic through the use of chaos and fractals. Now, with a simple formula, a computer can create a beautiful, and realistic tree. Instead of following a regular pattern, the bark of a tree can be created according to a formula that almost, but not quite, repeats itself.

Music can be created using fractals as well. Using the Lorenz attractor, Diana S. Dabby, a graduate student in electrical engineering at the Massachusetts Institute of Technology, has created variations of musical themes. ("Bach to Chaos: Chaotic Variations on a Classical Theme", Science News, Dec. 24, 1994) By associating the musical notes of a piece of music like Bach's Prelude in C with the x coordinates of the Lorenz attractor, and running a computer program, she has created variations of the theme of the song. Many musicians who hear the new sounds believe that the variations are very musical and creative.

The philosophical implications of chaos have already had a lasting effect on science. Now it is recognized that even when it is possible to reduce a system to a simple mathematical model, the question is more complex than if the system has a stable or unstable solution. Now there's something in between: the chaotic attractor, a bit more stable than complete randomness, but more dynamic than a simple periodic system.


References


The first draft of this essay was for my final project in a high school advanced composition class. It has been available online at my web site since 1996, most recently at the address http://www.imho.com/grae/chaos (and that one has pictures...). This version has been edited for use on E2.

  • "Bach to Chaos: Chaotic Variations on a Classical Theme", Science News, Dec. 24, 1994, pg. 428.
  • Gleick, James, Chaos - Making a New Science, Penguin Books Ltd, Harmondsworth, Middlesex, 1987.
  • Lowrie, Peter, personal interview over the Internet, May 17, 1995.
  • Rae, Kevin, "Chaos", unpublished paper, submitted to Professor Gould, Modern Physics class, Claremont McKenna College, December 5, 1994.
  • Stewart, Ian, Does God Play Dice? The Mathematics of Chaos, Penguin Books Ltd, Harmondsworth, Middlesex, 1989.

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