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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 theMath G Projects Page.

This paper received a Study WebAward.
 
 

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

Instead I will be writing about Chaos Theory. I haveheard a lot about this type of math mainly on TV; I've also read bookslike "Jurassic Park" which uses it as an example of what happens when manmesses with nature. I have also read a little about it in news magazineslike "Newsweek" and learned about how scientists are using this somewhatnew form of math to help predict events. I will do a basic overview ofChaos Theory and also give a history of where it came from and who designedit. I will demonstrate how it is used in everyday life, and discuss someof 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 thecharacter 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 theheart are examples of things that can be described by non-linear equations.Linear equations are, according to Dr. Malcom, hard if not impossible tosolve. So the new theory that would describe these events is called "ChaosTheory".

Dr Malcom says chaos is not all random theory, thereare hidden regularities within the complex variety of the system's behavior.He says Chaos Theory is based on two things: one, complex systems likeweather have an underlying order; and two, (reverse of the first) simplesystems can produce complex behavior. Thus Chaos Theory can be used tostudy the stock market, rioting crowds and brain waves during an epilepticseizure.

As for the history of Chaos Theory, it is relativelya brief one. Chaos Theory was invented in 1960 as part of an attempt tomake computer weather models. Chaos Theory is a concept discovered by ameteorologist named Edward Lorenz. Lorenz was working on a computer programthat would help predict weather. What he would come up with would not predictthe weather itself, but it would predict what the weather might be. Whileworking on this predicting program, he came across an unusual find. Hewas testing to see if he could get a particular number sequence for a secondtime. He started in the middle of the first number sequence. After lettingthe computer print out the information for about an hour, he discoveredthat the sequence had evolved differently. The product was a pattern thatended up totally different from the first. He would figure out what hadhappened, to save paper on his project he had it only print out three decimalplaces. Lorenz's original number was .506127 but he had only typed thefirst three digits which were .506. According to the article I found onthe internet about Chaos, his experiment should have worked, meaning heshould have ended up with a sequence close to the original. What Lorenzhad come up with from this was the butterfly effect. This means that theamount of difference in the starting points of the two curves is so smallthat specialists compare it to a butterfly flapping its wings. This Webarticle quoted Ian Stewart's book "Does God Play Dice - The Mathematicsof Chaos."  When Stewart says of the butterfly effect:

"The flapping of a single butterfly's wing todayproduces a tiny change in the state of the atmosphere. Over a period oftime, what the atmosphere actually does diverges from what it would havedone. So. in a month's time, a tornado that would have devastated the Indonesiancoast doesn't happen. Or maybe one that wasn't going to happen does."

This is an excellent example that would help explainto me this concept of Chaos. It maps out in a simple phrase how powerfulthis theory can be summed up by just using words; however, the math partis the real trick. I will get more into that later in this paper. The articlestates that this example is known as sensitive dependence on initial conditions.The slightest change in the starting conditions can drastically changethe behavior or outcome of an entire system. From this Lorenz stated thatit is impossible to accurately predict the weather (not news to the mostof us who pay attention to today's weather reports! !). This outcome broughtLorenz to look into more aspects of this type of math that would be knownas Chaos Theory. Lorenz would start to dedicate his work to looking foran 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 breakdown to three equations that did have sensitive dependence on their initialconditions. He would later discover that the three equations he inventedwould describe a water wheel. This is summed up by James Gleick's bookcalled "Chaos - Making A New Science." Gleick says:

"At the top, water drips steadily into containershanging on the wheel's rim. Each container drips steadily into a smallhole. If the stream of water is slow, the top containers never fill fastenough to overcome friction, but if the stream is faster, the weight startsto turn the wheel. The rotation might become continuous. Or if the streamis so fast that the heavy containers swing all the way around the bottomand 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 meteorologicalone and not a math one, because he was not a mathematician. It would takeyears for Lorenz work to be discovered by the math world and when the mathspecialists heard about it, they realized that Lorenz had discovered somethingrevolutionary.

Today, Chaos theory is widely used in both the scientificrealm, as well as our everyday lives, and we still don't really noticeit is being used! One  use for Chaos Theory is in the study in thedaily operations of the stock market. According to an internet articleon 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 typeof non-linear system that the stock market is. This theory has determinedthat prices in the stock market are highly random and are random in a trendyway. The trends are dependent on the time you are looking at the stockmarket because they vary. Another concept involved in chaotic systems isfractals. Fractals can be described as parts that are considered to be"self similar" in that they are related to a whole. This internet articlegives the example of a tree as an example of a fractal system. While thebranches get smaller, each is similar in structure to the larger branchesand the tree as a whole. To relate to this example, you look at the stockmarket price action. On  monthly, weekly, daily bar charts, thesestructures have similar characteristics. This article goes on to statethat even if we could predict tomorrow's stock market change exactly (whichis impossible), we would still have zero accuracy in trying to predictonly twenty days ahead.

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

Another everyday use of Chaos Theory is in the conceptof weather predicting. Predicting what the weather is going to do overthe next three or four days is not so difficult to do on a basis whereyou use accuracy of the predictions. But, if you want to predict what theweather 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 pressureand more are used to describe what the weather is doing (this accordingto another part of an article found on the internet on Chaos Theory). Equationsthat control the weather involve all these things. Edward Lorenz triedto put all the weather variables into an equation and calculate the degreeof certainty of the value of all the variables. He let his computer dothis for several days ( I discussed this in the first part of this paper).We already know what happened after this when he shortened the equationto three decimal places. Lorenz discovered for the science of meteorologythat we can't measure the weather variables accurately enough to avoidthe effects of chaos.

Science uses Chaos Theory to help describe many ofthe aspects of the universe we live in. Not only is it used in the scienceof predicting weather. but it is also used to describe how our solar systemworks for example. To astronomers, Chaos theory is not new ( this accordingto 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 changesoccur in places like, the motion of Saturn's moon Heparin, the gaps inthe asteroid belt between Mars and Jupiter, and in the planets' orbits.For example, an object can be orbiting in a certain way for thousands ormaybe millions of years, then in an instant change, thus changing the object'sfuture and making its past irrelevant to that future. One example of howChaos is prevalent in astronomy is astronomers can easily predict how anytwo bodies will travel around thei common center of gravity. An examplegiven in this internet article is, the Earth and the moon. they travelaround a third form of gravity which is the sun. The sun prevents a definitiveanalytical solution to the equations of motion. This is what makes thelong term evolution of the system impossible to predict. Even with allthe computers and calculators that are so high tech today, it is stillimpossible to keep up with the pace of chaos (which is one of the mainreason it is called chaos because it can't be controlled or predicted).

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

"Considerable historical evidence indicates thatscientific progress is the exclusive outcome of neither the empirical northe philosophical domains; rather, great progress in the scientific disciplinesresults 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 ofmetaphysics and empiricism, you can always come up with correct solutionsto scientific questions. This is because there is no third "chaotic" variableto make the study chaotic, like in the study of the solar system and weatherexamples where there was plenty of other variables to cause chaos. Withthese two variables in science, you can always have a correct analysisor prediction, but you have to have both of these converge to make thescience correct. That's what I got from this article after analyzing itand 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 intothis type of math? I have found several examples on the internet that helpdemonstrate what kind of math is used in solving chaotic problems. Someof these examples are almost self explanatory, while some are much morecomplex.

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

What Did I Do ?

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

2) I then erased the middle third of the line (between1/3 and 2/3) and was left with two thirds of the original line.

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

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

Through infinite iteration, you will eventually endup with a set of points that remain in the Cantor Set. A graph of one ofthese sets would represent "dust like" scattered, unconnected points.Itturns out that when certain functions are iterated many times they willproduce effects that are similar to the Cantor Set. That's what this examplehas to do with Chaos.

The Cantor Set is just one set in the many that makeup Chaos type of math. One other such type of thought is called "The ComplexPlane". The coordinates in which the x-axis measures the real part of anumber, and the y-axis measures the imaginary part, which is symbolizedas 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 partof the Mandelbrot set is Complex Functions. This, simply put, maps a complexvalue to another complex value. This is similar to the real function thatmaps a real number to another real number. The difference here is the imaginarypart of the complex number cannot mix with the real part. My source onthis compares this to apples and oranges, which are two parts of the samevalue that must always be considered seperatly. My source cites an excellentexample of how this function works. This chart compares the outcomes ofvarious 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 twosets are part of: the Mandelbrot Set. This set is similar to the CantorSet in the way that both sets consist of points that did not escape wheniterated through a function. The difference here is this set is obtainedusing the complex function: f(z)=z²+c, where z is the complexindependent variable and c is the complex parameter. The fascinating pointin the Mandelbrot Set is the actual points of the set did not escape ordiverge from infinity.

Another set function is the Julia Set. This set usesthe function f(z) = z²+c, where z is the complex independentvariable, and c is the complex parameter that is constant throughout theset. The thing that makes the Julia Set so interesting is if you pick aconstant parameter that is inside the Mandelbrot Set. the Julia set willbe connected! However, if you pick a constant parameter that is outsideof 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 intendedfor the general public; therefore, it should have been fairly understandableto someone like me. I felt I understood a better part of the math, howeverI did have to wade through a lot of stuff that was really over my headas far as my experience with regular math goes! But that brings me to aconclusion: the study of Chaos and Chaos Theory is an extremely difficultone. I have learned that these non-linear equations are very complex andimportant stuff. It seems to me that if we can somehow learn how to getpast the problems we have with non-linear equations, we can solve someof the greatest mysteries that science and the universe has to offer. Ifanything, my experience in looking at sources for this paper had reallyenlightened me to the importance of this type of math. Yet when I put allthe elements together of what chaos is and I look at what  kind ofmath and thinking goes in to chaos, I realize how the term Chaos is theperfect one in describing what happens in all the elements that make upchaos theory.
 
 

BIBLIOGRAPHY

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

Application of Chaos Theory to Psychological Modelshttp://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/capecanavral/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.