Math
Department, Mission College, Santa Clara, California
Midterm Paper on Chaos
Dusty Sykes
Math G, Spring 04
This subject has moved from interesting and relatively comprehensible to me after viewing the video to relatively impenetrable and mystifying after reading a great deal about it in reference books over the last week and one-half. Chaos is said to be one of the three ìBig Ideasî of the 20th Century, the other two being quantum mechanics and Einsteinís work. So, I guess it should not be too surprising that I am not able to grasp it after a couple of weeks of intermittent effort. I keep feeling that there will be a ìbreak throughî that will permit my mind to grasp the concepts, but so far it hasnít happened. I started, inadvertently, with some of the denser, less easy references and then moved on to some of the more understandable. But even then, the number of metaphors and examples of the different parts of the subject overwhelmed me. It brought me into realms that were intriguing and of significant interest to me, but which were very large spaces. The elastic planes of Poincare, phase mapping, and many others are large concepts that require time to absorb and reflect on.
This report will be in the form of an inverted pyramid, with a report on the resources I used and ending with whatever synopsis I can arrive at. Perhaps the writing of this report will be the event that creates the breakthroughóweíll see. The details of the references are listed following the report, so I will only use title references in the body of the report.
The most accessible book for me was the Turbulent Mirror. It used Alice in Wonderland and a Chinese Philosopher, Chuang Tzu, as guides to the material. On a philosophical basis, the authors showed how human myths and religions treated chaos and how they sought to understand chaos through denial, acceptance and transformation. The message I took away is how powerful the desire for order is in humans and how that has corrupted scientific thought at timesóan orderly world was selected over a disorderly world, and excuses were made for the disorder. The bookís central message is that order and chaos coexist as part of a greater whole and both are to be embraced and understood.
I was introduced to the notion of phase space (page 35), which is a central idea in describing the behavior of many non-linear equations. The plot of a pendulum, with the y axis being position and the x axis being momentum was used. I can see where that is a useful notion, but it is a concept that I can understand in the moment, put eludes me when I encounter it later on. Discussions of fixed point attractors, limit cycle attractors, and saddles are much more understandable than others I encountered. Interesting examples are used to describe eachópike and trout (predator and prey). Introducing fisherman and adding another dimension made it much more difficult to understand. Discussions of the torus as an attractor were elusive for meókinda made sense and then I got lost.
This book had an entire chapter on strange attractors, which I had hoped would help me break the code and understand them, but the concept still escapes me. A useful analogy of water in a stream whose flow is interrupted by a rock was very helpful in general in understanding the stages of behavior. The message I took from this is that the amount of energy in a system is a major factor of whether or not the system will encounter chaos. The notion of subdividing also poked its head up, but that is also proving to be an elusive notion for me.
All the books that I read, including this, have introduced me to a great many mathematicians who have worked in areas related to turbulence. This has been of more than passing interest to me. Very interesting to see how many minds can be working on issues and contribute to an overall understanding of an issue. Leonardo drew pictures of turbulence, by subdividing vortexes, Landau and Hopf followed on.
The notion of fractals keeps appearing and it too is elusive. On page 51, the metaphor of crumpling a piece of paper is used. ìThe more tightly itís compressed, the more chaotic are its folds, and the closer the two-dimensional surface moves to becoming a three-dimensional solid. The Benard convection is like the crumpling paper, or a science fantasy character unable to choose between worlds. In a desperate ìeffortî to escape to a higher dimension or return to a lower one, the current wanders in the infinite byways of ìindecisionî between the two dimensions and thus crumples up. The dimension that this ìindecisionî inhabits is therefore not a whole dimension (not two-dimensional or three-dimensional) but a fractional dimension. And the shape the indecision traces is a strange attractor.î [This is dense writing. One could chew on this paragraph for a long time.]
A very interesting chapter used a formula for birthrates with a limiter formula to show that a point of stability was reached for a wide range of birthrates. It also showed that for some birthrates there was a bifurcation and even multiple bifurcations which put the system into chaos. He mentioned normalization, which I had encountered in a consulting project in which some database experts kept mentioning normalizing. It was nice to see normalization explained, however briefly.
The notion of intermittency was introduced and related to a diagram on page 31 which was extremely rich in detail and would be suitable for hours and hours of analysis. An example of system generated static in a system was used and the notion that massive networks were prone to intermittent issues was interesting.
In this book and others the iterating of a formula was most interesting. Issues such as periodicity were introduced. On page 72, a computer simulation of Poincareís picture being folded and after 10 iterations, it was a series of black and white lines, after 48 iterations there were several images of Poincare and after 241 iterations, the original picture of Poincare appeared. The impact of rounding errors was explored and was shown to have great impact on iterationsóleading to chaos by accumulation of rounding errors. An example of a universe-sized computer, rounding to 31 places would produce chaos in an iterative example after only 100 iterations. ìAt the speed which normal computers do iterations, predictability vanishes within a fraction of a second when highly nonlinear equations are concerned.î [Emphasis is mine.]
In Chapter Four, the story of John Russell, who came across a solitionóa solid wave that moved for several miles without dissipating is told. This is an example of chaos holding things together. The wave formed from the coupling of many smaller waves. The speed of solitions are dependent on their heightóa tall thin wave could catch up with a short fat one. An interesting idea was put forwardóthe sound of a distant cannon will always be heard before the order to fire. This is because the sound of the cannon travels as a solition. Itís interesting that Russell spent his entire life studying this phenomenon. Tsunamis are solitions.
David Ruelleís Change and Chaos is a remarkably well written book, and at the beginning appears to be quite approachable, but soon gets into areas that were beyond my ability to grasp. I think that the authors of all the books that I read were so immersed in their subject for so many years and for so much of their working life that they just can not break down the subject any other way. Itís such a large field of so much complexity, that it can only be described to a certain level of simplicity. After that, one needs a good sized ìtool kitî of knowledge and insight. Ruelle is French and has a beautiful command of the language. He also has a Puckish sense of humor.
Some examples:
From Acknowledgements: ìArthur Wightman and Laura Kang Ward fought nobly in defense of the English language.î
From the Preface: ìSpeaking of scientific colleagues, some of them will be upset by my unglorious descriptions of scientists and the world of research. For this I offer no apology: if science is the research of truth, should one not also be truthful about how science is made.î
From Chapter One: ìÖwe shall try to understand something of the triangular relation between the strangeness of mathematics, the strangeness of the physical world, and the strangeness of our own human mind.î
This book is as much a work of philosophy as it is one of math.
Does God Play Dice? was an approachable book initially. It defines chaos as ìStochastic behavior occurring in a deterministic system.î Further, ìÖthe definition is a paradox. Deterministic behavior is ruled by exact and unbreakable law. Stochastic behavior is the opposite: lawless and irregular, governed by chance. So chaos is ëlawless behaviour governed entirely by law.í ì
Stewart mentions calculator chaos. Using a calculator and iterating, some interesting patterns are seen. Iterating x squared for a value less than one, yields a limit of zero within 9 iterations. Using .54321 and pressing cosine button in the radian mode about 40 times, the number .739085133 appears and does not change for successive iterations. Both are examples of convergence.
He iterated a tangent function 300,000 times and it never converged nor turned periodic, although it increased very slowly at timesóby .0000001 per iteration. This is an example of intermittency, which is a typical behavior.
The square root button converges on 1. The 1/x button eventually alternates between .54321 and 1.840908673óan example of periodicity. Iterating the equation of 2x squared-1 with values of .054321 and .054322 looks very similar for awhile, but after the 50th iteration, it is completely different.
An example of kx squared ñ 1 shows that for k=1.4, there is a complicated cycling amongst 16 different values. Chaos sets in at about k = 1.5 and gets worse as k is increased. But with chaos fully developed at k = 1.74, at k = 1.75, there is a cycle with three numbers--.744, -.03 and -.998.
He goes into an interesting explanation of the solar system and tells of a Greek machine with gears that emulated the solar system, using the notion of epicycles. He tells of the evolution of thinking about the solar system.
Under the topic, The Reformulation Period, he uses a nice turn of phrase: ì In 1750 Lagrange took up Eulerís ideas and produced from them an elegant and far-reaching reformulation of dynamics. Two important ideas crystallized out of his work. Both had been around, as half-baked ideas, for decades, but Lagrange baked them golden brown, took them out of the oven, and placed them on the bakery counter for all to admire, buy and consume.î [Emphasis is mine.] Hamilton refined this further to arrive at a single quantityóthe Hamiltonian which defines the total energy in terms of positions and momentums.
On pages 54 and 55, he mentions the state of scientific thinking at the beginning of the 20th Century. ìAs the 20th Century unrolled, statistical methodology took its place alongside deterministic modeling as an equal partner. A new word was coined to reflect the realization that even chance has its laws: stochastic. (The Greek word stochastikos means ëskillful in aimingíÖ.. No longer was order synonymous with law, and disorder with lawlessness. Both order and disorder had laws. One law for the ordered, another for the disordered. Two paradigms, two techniques. Two ways to view the world. Two mathematical ideologies, each applying only within its own sphere of influence. Ö The two paradigms were equal partnersÖ. Equal. But different. Totally, irreconcilably different. Scientists knew they were different, and they new whyÖ. Between simplicity and complexity there could be no common ground. But what one generation of scientist knows, beyond any shadow of doubt, with a knowledge that is built into the very fabric of their world, is precisely what the succeeding generation will challenge and overturn. If you know something that strongly, you donít question it. If you donít question it, youíre living by faith, not by science.î [Bold and underlined emphasis is mine.]
This idea of being ìtrappedî by oneís knowledge is becoming a central idea of what Iím picking up in this course. Extending it out beyond math is interesting. This morning I read a page one article about Condeleeza Rice in the New York Times. It pointed out that her knowledge was based on the theory of super-power relationships and that she was not equipped to consider a stateless threat such as terrorism as being significant.
In describing Poincare, he says, ìThe voice of a man who touched chaosÖ And was horrified by it.î He also states that Poincare ìwas perhaps the last mathematician able to roam at will throughout every nook and cranny of his subject.î (btwóa description of the foldable picture of Poincare is also shown.) ìHe gazed into the abyss of chaos, he discerned some of the forms that lurked within; but the abyss was still dark and he mistook for monstrosities some of the most beautiful things in mathematics. Poincare had the depth, but he lacked the means of illuminati. It took another age, armed with Poincareís own qualitative theory of differential equations, together with computers and other technical assistance, to shine some light into the chaotic depths and reveal that beauty. But they could never done it if Poincare hadnít pioneered the way to the abyssís edge.î
Poincare developed the field of topology, also characterized as ërubber sheet geometry.í This was very mind stretching for me. Transformations from squares to circles and back and to a topologist all shapes are one. (quoted). Poincare had the notion of a plane intersecting a topological spaceócalled his surface of section. In a very involved way, he could use this to show periodicity.
A quote on non-linear equations. ìClassical mathematics concentrated on linear equations for a sound pragmatic reason: it couldnít solve anything else. In comparison to the unruly hooligan antics of a typical differential equation, linear ones are a bunch of choirboys. Ö So docile are linear equations that the classical mathematicians were willing to compromise their physics to get them. Ö. So ingrained became the linear habit that by the 1940s and 1950s many scientists and engineers knew little else. ëGod would not be so unkind, ë said a prominent engineer, ëas to make the equations of nature nonlinear.í ì [Emphasis mine.]
ìReally the whole language in which the discussion is conducted is topsy-turvy. To call a general differential equation ënonlinearí is rather like calling zoology ënonpachydermologyí. But you see, we live in a world which for centuries acted as if the only animal in existence was the elephant, which assumed that the holes in the skirting board must be made by tiny elephants, which saw the soaring eagle as a wing-eared Dumbo, the tiger as an elephant with a rather short trunk and stripes, and whose taxonomists resorted to corrective surgery so that the museumís zoological collection consisted entirely of lumbering grey pachyderms. So ënonlinearí it is.î
Pages 95 to 125 are devoted to the chapter entitled ìStrange Attractors.î I was really stoked to see this, as I knew that this was a key to an understanding of chaos. However, it reached beyond my ability to understand and digest in the short period of this project. There is an interesting diagram and discussion of sinks, saddles, limit cycles and sources which ìtypicallyî are the 4 conditions met in a system of differential equations. Next is a discussion of quasiperiodic motions that take place on a torus, tied together by the idea of swinging a cat in space. Then a discussion of Smaleís ideas of attractors, and Poincareís sections and tenfold wrapping. My head hurt lots here.
And finally ending up with ìCantor Cheese.î This is a problem that starts with one line and removes the middle third, resulting in two lines with a space equal to their length in between them. Now iterate until you have LOTS of fragments. Points such as ‡, 1/9, 2/9,7/9, and 8/9 escape removal. (quote) Anyway, I have seen many versions of the Cantor set, but really donít ìgetî what itís going after.
This book went on to more difficult topics, but I gave up after about 125 pages.
I was particularly disappointed in The Essence of Chaos by the ìFather of Chaosî, Edward Lorenz. I read 75 pages of which most was a discussion of cross slope differences, using a skiing analogy with moguls. I think there might be more of interest later in the book, but I did not enjoy the writing style.
I scanned Newtonís Clock and was not up to reading more about chaos, particularly when it was focused on the solar system, which is a topic that has eluded me for sometime and for which I had no further energy.
Symmetry in Chaos is a visually stunning book. I didnít get much of the math in it. I have a better ideal of some of the types of symmetry: rotational, flip and others. The images in this book are amazing to me. Tiling, wallpapers, fractal images, geometric, and on and on. One interesting thing was the number of iterations of some of these imagesó30 million iterations of an equation were required to generate images. The colors were developed by the numbers of ìhitsî on a pixel during all the iterations. Like I say, I really donít understand it, but nonetheless, I am quite impressed with the results.
Web Sites
I am ìcutting and pastingî from these sites, for info that fills in some of the holes from the books that I read. My original thoughts are in bold type.
http://en.wikiphttp://users.ox.ac.uk/~quee0818/chaos/chaos.htmledia.org/wiki/Chaos_theory
This site has hundreds of links and it provides quite a technical explanation of chaos. Not sure that an average person would get much out of it.
http://users.ox.ac.uk/~quee0818/chaos/chaos.html
An interactive view of chaos, but not for the ordinary student, although it supposedly is aimed at undergraduate science majors.
«ambel (1993) defines chaotic systems as "[s]ystems that upon analysis are found to be nonlinear, nonequilibrium, deterministic, dynamic, and that incorporate randomness so that they are sensitive to initial conditions, and have strange attractors".
This a pretty random source of information and there is no overall plan to the presentation of the story of Chaos. The following quote was interesting nonetheless.
From Poincare:
ìÖBut it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. - in a 1903 essay "Science and Method" ì
http://www.zeuscat.com/andrew/chaos/chaos.html
This site is about 8 years old and no longer added to. It did have some interesting ìfactoidsî, which are included, as well as some droll humor at the end.
The Lorenzian Waterwheel
Most casual armchair scientists have no access to uniformly smooth boxes and elemental gases, much less instruments to measure the rotational speed of a moving cylinder of gas.
A metaphor for the gaseous system is found in the Lorenzian waterwheel. This is
a thought experiment. Imagine a waterwheel, with an arbitrary number of
buckets, usually more than seven, spaced equally around its rim. The buckets
are mounted on swivels, much like Ferris-wheel seats, so that the buckets will
always open upwards. At the bottom of each bucket is a small hole. The entire
waterwheel system is then mounted under a waterspout.
The scenario is set: now we commence the action.
Begin the flow of water from the waterspout. At low speeds, the water will
trickle into the top bucket, and immediately trickle out through the hole in
the bottom. Nothing happens. Real boring. Increase the flow just a bit,
however, and the waterwheel will begin to revolve as the buckets fill up faster
than they can empty. The heavier buckets containing more water let water out as
they descend, and when the water is gone, the now-light buckets ascend on the
other side, ultimately to be refilled. The system is in a steady state; the
wheel will, like a waterwheel mounted on a stream and hooked to a grindstone,
continue to spin at a fairly constant rate.
But even this simple system, sans boxes
or heated gases, exhibits chaotic motion. Increase the flow of water, and strange
things will happen. The waterwheel will revolve in one direction as before, and
then suddenly jerk about and revolve in the other direction. The conditions of
the buckets filling and emptying will no longer be so synchronous as to
facilitate just simple rotation; chaos has taken over.
What's a Fractal?
The Sierpinski Triangle raises all sorts of little questions that relate to
topics in chaos theory not covered in the last few pages. For example, the
Sierpinski Triangle is a canonical example of a shape known as a fractal. But soft, you ask, pray tell, what is a fractal?
Most simply, a fractal is a geometric construction that is self-similar at
different scales. This is rather dry. More clearly, a fractal shape will look
almost, or even exactly, the same no matter what size it is viewed at.
This is a pretty unintuitive concept. But let us look at the Sierpinski
Triangle. The first step in the geometric construction of the Sierpinski
Triangle involved splitting a triangle up into three other triangles. When we
look at the finished Sierpinski Triangle, we can zoom in on any of these three
sub-triangles, and it will look exactly like the entire Sierpinski Triangle
itself. In fact, we can zoom in to any depth we would like, and always find an exact replica of the Sierpinski Triangle.
This is deep. This is very deep.
Geometric Construction
The most conceptually simple way of generating the Sierpinski Triangle is to
begin with a (usually, but not necessarily, equilateral) triangle (first figure
below). Connect the midpoints of each side to form four separate triangles, and
cut out the triangle in the center (second figure). For each of the three
remaining triangles, perform this same act (third figure). Iterate infinitely
(final figure).
![[Sierpinski Construction]](sykes_files/image002.gif)
The result, as you can see, is the Sierpinski triangle. The geometric construction of the Sierpinski triangle is the most intuitive way to generate this fascinating fractal; however, it is only the tip of the Sierpinski iceberg.
As a total aside, I have found that methodically drawing the Sierpinski triangle during boring lectures greatly relieves stress. If more boring lectures are anticipated, draw a huge one, like one that spans an entire sheet of regular paper drawn to painstaking detail. After a few lectures your boredom will be greatly relieved, your stress will go down, chicks (or hunks, as the case may be) will dig you, and you'll end up with a really, really impressively detailed (and large) Sierpinski Triangle which people will be really impressed with. They will say things like "Man, that's cool!" and "Whoa, how'dja do that!" and "Man, you must have been really bored."
http://www.duke.edu/~mjd/chaos/chaosh.html
This is an undergraduate
paper.
In addition to the famous
Sierpenski Triangle, the Koch Snowflake is also a well noted, simple fractal image.
To construct a Koch Snowflake, begin with a triangle with sides of length 1. At
the middle of each side, add a new triangle one-third the size; and repeat this
process for an infinite amount of iterations. The length of the boundary is 3 X
4/3 X 4/3 X 4/3...-infinity. However, the area remains less than the area of a
circle drawn around the original triangle. What this means is that an
infinitely long line surrounds a finite area. The end construction of a Koch
Snowflake resembles the coastline of a shore.
On February 3rd, 1893, Gaston
Maurice Julia was born in Sidi Bel Abbes, Algeria. Julia was injured while
fighting in World War I and was forced to wear a leather strap across his face
for the rest of his life in order to protect and cover his injury. he spent a
large majority of his life in hospitals; therefore, a lot of his mathematical
research took place in the hospital. At the age of 25, Julia published a 199 page
masterpiece entitled "Memoire sur l'iteration des fonctions." The
paper dealt with the iteration of a rational function. With the publication of
this paper came his claim to fame. Julia spent his life studying the iteration
of polynomials and rational functions.
The Cantor set is simply the
dust of points that remain. The number of these points are infinite, but their
total length is zero. Mandelbrot saw the Cantor set as a model for the
occurrence of errors in an electronic transmission line. Engineers saw periods
of errorless transmission, mixed with periods when errors would come in gusts.
When these gusts of errors were analyzed, it was determined that they contained
error-free periods within them. As the transmissions were analyzed to smaller
and smaller degrees, it was determined that such dusts, as in the Cantor Dust,
were indispensable in modeling intermittency. This answers one of my questions, but all the
implications are still ëout there.í
http://www.hypertextbook.com/chaos/
This is an interesting site. Not for the timid. Pretty heavy duty math and dense discussions. Discussion of Dimensions and how it applies to Topology and Fractals are interesting, but not easy.
An excellent bibliography with lots of links to other web sites.
http://mathforum.org/library/topics/chaos/
This site is from Drexel University and has links to 155 different sites about chaos or fractals
http://library.thinkquest.org/3703/
This is a site that focuses more on fractals. It has down loadable programs to create fractal images. It also has a Predator-Prey Model and a Game of Life that both looked interesting.
Chaos is
apparently unpredictable behavior arising in a deterministic system because of
great sensitivity to initial conditions. Chaos arises in a dynamical system if
two arbitrarily close starting points diverge exponentially, so that their
future behavior is eventually unpredictable.
What this means is that chaotic behavior, although appearing random, arises
from a very rigid cause. It also is highly sensitive to any disturbances, because
every change in the system will compound with time. Also, because of the
extreme disorder, predicting the future path of the system is practically
impossible.
http://library.thinkquest.org/3120/
This site was put up by a group of high school students. It is a testimony to how much kids in HS can do, but was not of much interest to me otherwise.
http://order.ph.utexas.edu/chaos/index.html
This is a ìniceî interactive explanation with virtually
no math, that gives a very broad view of the history and ideas of chaos theory.
SUMMARY
I think that I have learned a great deal about the human side of how knowledge about the world progresses along its uneven course. The common theory that it builds on the past is not as ìtrueî as a picture that has many discontinuities, many of which are discovered or developed by a single genius. This genius is not beholden to the past and has a keenly questioning mind. Also a mind of extraordinary capaciousness or focus along a given line of inquiry. These new ideas are often met with resistance and/or ridicule by those whose fame and fortune is tied to previous ideas.
I have increased the scope of my thinking considerably during the process of writing this paper. I am intrigued by the wholeness of thingsóthe ability of nature (and of mankind) to hold opposites simultaneously. It has been interesting to see how uncomfortable it has been for mankind to think of the world as having a chaotic element to it and how earnestly so many have worked to find order. Sometimes the attempt has been to exert order where it does not exist. This seems to have its analog in psychologyóour ìdarkî sides are seen as disorders and not part of our total personalities and other realms of human endeavor that seek to winnow out the bad/imperfect and leave only the ìgoodî behind.
The insight about non-linear being similar to non-pachyderm was most useful to me.
I think that I am more at ease with my ìunknowingî about the details of chaos. I am now more willing to be happy about having been exposed to so much and to know that there is a different universe for me to explore. One that will be fun, interesting and challenging for me.
Thanks for exposing me to the Nova film. After having completed this project, it is more and more apparent what an excellent job of presenting the topic of chaos that they achieved.
References from Books
Field, Michael, and Martin Golubitsky. Symmetry in Chaos. Oxford: Oxford University Press, 1992.
Briggs, John, and F. David Peat Turbulent Mirror. New York: Perennial Library, Harper & Row, 1989.
Stewart, Ian. Does God Play Dice? The Mathematics of Chaos. Cambridge, MA:
Basil Blackwell Inc., 1989.
Ruelle, David. Chance and Chaos. Princeton, N.J.: Princeton University Press, 1991
Peterson, Ivars. Newtonís Clock Chaos in the Solar System. New York: W.H. Freeman and Company, 1993.
Lorenz, Eward N. The Essence of Chaos. Seattle: University of Washington Press, 1993.
References from Periodicals
Iannone, Ron. ìChaos theory and its implications for curriculum and teaching.î Education Summer 1995, v115 n4 pgs 541 to 547
References from the www
There were 1,660,000 references called up by Google on the search ìchaos theoryî and 457,000 references on the search ìchaos theory basics.î Here are some of the more interesting ones that I plan to use.
Rae, Gregory. Chaos Theory. Self published http://www.imho.com/grae/chaos/index.html accessed on 3/22/2004
The Math Forum at Drexel University (Philadelphia) This site provides links to over 150 other sites on chaos http://mathforum.org/library/topics/chaos/ on 3/22/2004
Trump, Mathew A. The University of Texas What is Chaos? A five-part online course for everyone http://order.ph.utexas.edu/chaos/index.html accessed on 3/22/2004
This site has a tutorial on Chaos. The five parts are: The Philosophy of Determinism, Initial Conditions, Uncertainty of Measurements, Dynamical, Instabilities and Manifestations of Chaos. This work was done at the Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems. Prigogine was a 1977 Nobel Laureate in Chemistry who died last year.
Thinkquest Three high school students created this web site libraryhttp://www.thinkquest.org/library/site_sum.html?tname=3120&url=3120 accessed on 3/22/2004 ThinkQuest is an international competition where student teams engage in collaborative, project-based learning to create educational websites. The winning entries form the ThinkQuest online
Glenn Elert This looks like a very useful website. Not sure who Glenn Elert is, but he seems to have done a great job on this site. http://www.hypertextbook.com/chaos/ accessed on 3//22/2004.
Manus J. Donahue III Duke University This is an undergraduate paper that seems to be extremely well written and useful http://www.duke.edu/~mjd/chaos/chaosh.html accessed on 3/22/2
Ho, Andrew senior software engineer at Tellme Networks, Inc. Self published website http://www.zeuscat.com/andrew/chaos/chaos.html004 accessed on 3/22/2004
University of Maryland Website for its multidisciplinary work in Chaos Theory http://www-chaos.umd.edu accessed on 3/22/2004
Reil, Torsten University of
Oxford, Introduction to Chaotic
Systems http://users.ox.ac.uk/~quee0818/chaos/chaos.html
accessed on 3/22/2004
Wikipedia, The Free Encyclopedia http://en.wikipedia.org/wiki/Chaos_theory accessed on 3/24/20004
Other
resources noted but not used. From
library catalog at Mission College.
Applied Chaos TheoryóCombel, QA 172.SC45 at West Valley
Impact of Chaos on Science and SocietyóQ172.5 C45I45 at West Valley
Introduction to Chaotic Dynamics, Dwaney, QA 614.D48 at West Valley
Natures Chaos, TR 721.658 at Mission College
Nova Video at WVC AV Circulation Desk
ChaosóNew Science. Gleick, QA 172.5C45G54 at West Valley. This appears to be a seminal work, but fairly complex.
Chaos theory tamed Williams, QA 172.5C45W55 at West Valley
Chaotic Dynamics, An Introduction, Baker and Golub, QA862.P4B35 at Mission College
Chaotic Dynamics of non-linear relationships, Rasband, Q172.SC45 at Mission College
Complexity: Life on the edge of chaos Lewin, Roger B105.c473L48