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This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.

To explore other such papers go to the Math G Projects Page.

This paper received a Study Web Award.
 
 

To shift gears so to speak, I will not be doing anything this time about math anxiety.

Instead I will be writing about Chaos Theory. I have heard a lot about this type of math mainly on TV; I've also read books like "Jurassic Park" which uses it as an example of what happens when man messes with nature. I have also read a little about it in news magazines like "Newsweek" and learned about how scientists are using this somewhat new form of math to help predict events. I will do a basic overview of Chaos Theory and also give a history of where it came from and who designed it. I will demonstrate how it is used in everyday life, and discuss some of the methods of this new form of math.

I found an interesting web page on Chaos Theory, by famous Bay Area mathematician Ian Malcom. Malcom is best known for the character that is based on him in both the "Jurassic Park" movies. Dr. Malcom says simple concepts like turbulence, water coming out of a spout, air coming over an airplane wing, weather, and blood flowing through the heart are examples of things that can be described by non-linear equations. Linear equations are, according to Dr. Malcom, hard if not impossible to solve. So the new theory that would describe these events is called "Chaos Theory".

Dr Malcom says chaos is not all random theory, there are hidden regularities within the complex variety of the system's behavior. He says Chaos Theory is based on two things: one, complex systems like weather have an underlying order; and two, (reverse of the first) simple systems can produce complex behavior. Thus Chaos Theory can be used to study the stock market, rioting crowds and brain waves during an epileptic seizure.

As for the history of Chaos Theory, it is relatively a brief one. Chaos Theory was invented in 1960 as part of an attempt to make computer weather models. Chaos Theory is a concept discovered by a meteorologist named Edward Lorenz. Lorenz was working on a computer program that would help predict weather. What he would come up with would not predict the weather itself, but it would predict what the weather might be. While working on this predicting program, he came across an unusual find. He was testing to see if he could get a particular number sequence for a second time. He started in the middle of the first number sequence. After letting the computer print out the information for about an hour, he discovered that the sequence had evolved differently. The product was a pattern that ended up totally different from the first. He would figure out what had happened, to save paper on his project he had it only print out three decimal places. Lorenz's original number was .506127 but he had only typed the first three digits which were .506. According to the article I found on the internet about Chaos, his experiment should have worked, meaning he should have ended up with a sequence close to the original. What Lorenz had come up with from this was the butterfly effect. This means that the amount of difference in the starting points of the two curves is so small that specialists compare it to a butterfly flapping its wings. This Web article quoted Ian Stewart's book "Does God Play Dice - The Mathematics of Chaos."  When Stewart says of the butterfly effect:

"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."

This is an excellent example that would help explain to me this concept of Chaos. It maps out in a simple phrase how powerful this theory can be summed up by just using words; however, the math part is the real trick. I will get more into that later in this paper. The article states that this example is known as sensitive dependence on initial conditions. The slightest change in the starting conditions can drastically change the behavior or outcome of an entire system. From this Lorenz stated that it is impossible to accurately predict the weather (not news to the most of us who pay attention to today's weather reports! !). This outcome brought Lorenz to look into more aspects of this type of math that would be known as Chaos Theory. Lorenz would start to dedicate his work to looking for an easier way to find a system of sensitive dependence on initial conditions. He would come up with a 12 equation system that he would eventually break down to three equations that did have sensitive dependence on their initial conditions. He would later discover that the three equations he invented would describe a water wheel. This is summed up by James Gleick's book called "Chaos - Making A New Science." Gleick says:

"At the top, water drips steadily into containers hanging on the wheel's rim. Each container drips steadily into 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 slow, stop, and reverse its rotation, turning first one way and then the other."

Unfortunately, for Lorenz, since he was a meteorologist, the only journal he could publish his new math findings in was a meteorological one and not a math one, because he was not a mathematician. It would take years for Lorenz work to be discovered by the math world and when the math specialists heard about it, they realized that Lorenz had discovered something revolutionary.

Today, Chaos theory is widely used in both the scientific realm, as well as our everyday lives, and we still don't really notice it is being used! One  use for Chaos Theory is in the study in the daily operations of the stock market. According to an internet article on uses of Chaos theory, the stock market is a non-linear, dynamic system. It turns out that Chaos Theory is the math used for studying this type of non-linear system that the stock market is. This theory has determined that prices in the stock market are highly random and are random in a trendy way. The trends are dependent on the time you are looking at the stock market because they vary. Another concept involved in chaotic systems is fractals. Fractals can be described as parts that are considered to be "self similar" in that they are related to a whole. This internet article gives the example of a tree as an example of a fractal system. While the branches get smaller, each is similar in structure to the larger branches and the tree as a whole. To relate to this example, you look at the stock market price action. On  monthly, weekly, daily bar charts, these structures have similar characteristics. This article goes on to state that even if we could predict tomorrow's stock market change exactly (which is impossible), we would still have zero accuracy in trying to predict only twenty days ahead.

Many stock market experts use this theory to state that if people trade with an intra-day five minute bar chart, they are trading random noise and that they are wasting their time. However, expert traders also say that long term price action is not really random. Traders have a better chance for success if they trade on a weekly chart basis if they just follow the trends.

Another everyday use of Chaos Theory is in the concept of weather predicting. Predicting what the weather is going to do over the next three or four days is not so difficult to do on a basis where you use accuracy of the predictions. But, if you want to predict what the weather will be like on your birthday that's, let's say, three months off, chaos theory can't even give you an accurate number.

There are many variables that pertain to the weather. Temperature, air pressure, wind speed, wind direction, barometric pressure and more are used to describe what the weather is doing (this according to another part of an article found on the internet on Chaos Theory). Equations that control the weather involve all these things. Edward Lorenz tried to put all the weather variables into an equation and calculate the degree of certainty of the value of all the variables. He let his computer do this for several days ( I discussed this in the first part of this paper). We already know what happened after this when he shortened the equation to three decimal places. Lorenz discovered for the science of meteorology that we can't measure the weather variables accurately enough to avoid the effects of chaos.

Science uses Chaos Theory to help describe many of the aspects of the universe we live in. Not only is it used in the science of predicting weather. but it is also used to describe how our solar system works for example. To astronomers, Chaos theory is not new ( this according to the same web page on Chaos Theory). To those who study stars and planets, chaos means an abrupt change in some object's orbit. Some of these changes occur in places like, the motion of Saturn's moon Heparin, the gaps in the asteroid belt between Mars and Jupiter, and in the planets' orbits. For example, an object can be orbiting in a certain way for thousands or maybe millions of years, then in an instant change, thus changing the object's future and making its past irrelevant to that future. One example of how Chaos is prevalent in astronomy is astronomers can easily predict how any two bodies will travel around thei common center of gravity. An example given in this internet article is, the Earth and the moon. they travel around a third form of gravity which is the sun. The sun prevents a definitive analytical solution to the equations of motion. This is what makes the long term evolution of the system impossible to predict. Even with all the computers and calculators that are so high tech today, it is still impossible to keep up with the pace of chaos (which is one of the main reason it is called chaos because it can't be controlled or predicted).

One very difficult to understand article I found on the internet on Chaos Theory, attempted to make a psychological model of what Chaos Theory is to science. This article was, I assume, intended for those that study chaos and have a lot of experience in working with the terminology that goes along with this kind of study. I managed to get this quotation out of this article after analyzing it:

"Considerable historical evidence indicates that scientific progress is the exclusive outcome of neither the empirical nor the philosophical domains; rather, great progress in the scientific disciplines results when metaphysics and empiricism converge."

(statement taken from mathematician Zukav, in 1979); this quote is taken from the same internet page.

I believe this states that with the variables of metaphysics and empiricism, you can always come up with correct solutions to scientific questions. This is because there is no third "chaotic" variable to make the study chaotic, like in the study of the solar system and weather examples where there was plenty of other variables to cause chaos. With these two variables in science, you can always have a correct analysis or prediction, but you have to have both of these converge to make the science correct. That's what I got from this article after analyzing it and comparing it to other things I have read in preparation for this paper.

So, after discussing the history of Chaos Theory, and giving some examples of how it is used in both daily life and science, where does the math come in? What kind of formulas and equations go into this type of math? I have found several examples on the internet that help demonstrate what kind of math is used in solving chaotic problems. Some of these examples are almost self explanatory, while some are much more complex.

First, I will demonstrate the Cantor set, or it's proper name of the Cantor middle thirds set. This is an example of a fractal number line. It's a set on the interval between 0 and 1 on the number line. The construction of these lines is pretty simple. In this example, a line represents a set of numbers, and "removing a section" is analogous to taking out that part of the set. The following is a demonstration that I found on a Web page on Chaos math:

What Did I Do ?

I ) I drew a horizontal line and labeled the left and right endpoints 0 and 1, respectively. This line represents the interval of real numbers between 0 and 1.

2) I then erased the middle third of the line (between 1/3 and 2/3) and was left with two thirds of the original line.

3) Next, I erased the middle thirds from both of the new lines.

If you repeat this step a few times you will begin to see a pattern. (the pattern is actually a fractal!)

Through infinite iteration, you will eventually end up with a set of points that remain in the Cantor Set. A graph of one of these sets would represent "dust like" scattered, unconnected points.It turns out that when certain functions are iterated many times they will produce effects that are similar to the Cantor Set. That's what this example has to do with Chaos.

The Cantor Set is just one set in the many that make up Chaos type of math. One other such type of thought is called "The Complex Plane". The coordinates in which the x-axis measures the real part of a number, and the y-axis measures the imaginary part, which is symbolized as i, of a number is what the Complex Plane is.

The Complex Plane is part of an overall method titled "The Mandelbrot Set." I will get more into the specifics of this later. Now let me describe another function of the Mandelbrot Set. The next part of the Mandelbrot set is Complex Functions. This, simply put, maps a complex value to another complex value. This is similar to the real function that maps a real number to another real number. The difference here is the imaginary part of the complex number cannot mix with the real part. My source on this compares this to apples and oranges, which are two parts of the same value that must always be considered seperatly. My source cites an excellent example of how this function works. This chart compares the outcomes of various x-values from the function f(x):

f(x) =x² for all real and complex x values
 
 

x	Real value	Imaginary value	Real squared	x squared
4	4	0	4	4
2 + 2i	2	2	4	(2+2i)(2+2i) = 8i
3 + 2i	3	2	9	(3+2i)(3+2i) = 5+12i
6 + 3i	6	3	36	(6+3i)(6+3i) = 27+36i

This leads me up to the main part of what these two sets are part of: the Mandelbrot Set. This set is similar to the Cantor Set in the way that both sets consist of points that did not escape when iterated through a function. The difference here is this set is obtained using the complex function: f(z)=z²+c, where z is the complex independent variable and c is the complex parameter. The fascinating point in the Mandelbrot Set is the actual points of the set did not escape or diverge from infinity.

Another set function is the Julia Set. This set uses the function f(z) = z²+c, where z is the complex independent variable, and c is the complex parameter that is constant throughout the set. The thing that makes the Julia Set so interesting is if you pick a constant parameter that is inside the Mandelbrot Set. the Julia set will be connected! However, if you pick a constant parameter that is outside of the Mandelbrot set, the Julia set will be an unconnected set of "dust" like points.

That's the math aspect of Chaos that I looked at. These examples were found on the internet, on sites that were intended for the general public; therefore, it should have been fairly understandable to someone like me. I felt I understood a better part of the math, however I did have to wade through a lot of stuff that was really over my head as far as my experience with regular math goes! But that brings me to a conclusion: the study of Chaos and Chaos Theory is an extremely difficult one. I have learned that these non-linear equations are very complex and important stuff. It seems to me that if we can somehow learn how to get past the problems we have with non-linear equations, we can solve some of the greatest mysteries that science and the universe has to offer. If anything, my experience in looking at sources for this paper had really enlightened me to the importance of this type of math. Yet when I put all the elements together of what chaos is and I look at what  kind of math and thinking goes in to chaos, I realize how the term Chaos is the perfect one in describing what happens in all the elements that make up chaos theory.
 
 

BIBLIOGRAPHY

Chaos . a web page. http://tqd.advanced.org/2647/chaos.htm

Application of Chaos Theory to Psychological Models http://www.perfstrat.com

In A World of Order...Chaos Reigns! http://tqd.advanced.org/3120/main

Ian Malcom's Chaos Theory Homepage. www.geocities.com/cape canavral/6211/

This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.