Midterm Paper on Chaos

Dusty Sykes

Math G, Spring 04

This subject has moved from interesting and relativelycomprehensible to me after viewing the video to relatively impenetrable andmystifying after reading a great deal about it in reference books over the lastweek and one-half.  Chaos is saidto be one of the three “Big Ideas” of the 20^th Century, the othertwo being quantum mechanics and Einstein’s work.  So, I guess it should not be too surprising that I am notable to grasp it after a couple of weeks of intermittent effort.  I keep feeling that there will be a“break through” that will permit my mind to grasp the concepts, but so far ithasn’t happened.  I started,inadvertently, with some of the denser, less easy references and then moved onto some of the more understandable. But even then, the number of metaphors and examples of the differentparts of the subject overwhelmed me. It brought me into realms that were intriguing and of significantinterest to me, but which were very large spaces.  The elastic planes of Poincare, phase mapping, and many othersare large concepts that require time to absorb and reflect on.

This report will be in the form of an inverted pyramid, witha report on the resources I used and ending with whatever synopsis I can arriveat.  Perhaps the writing of thisreport will be the event that creates the breakthrough—we’ll see.  The details of the references arelisted following the report, so I will only use title references in the body ofthe report.

The most accessible book for me was the TurbulentMirror.  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 andreligions treated chaos and how they sought to understand chaos through denial,acceptance and transformation.  Themessage I took away is how powerful the desire for order is in humans and howthat has corrupted scientific thought at times—an orderly world was selectedover a disorderly world, and excuses were made for the disorder.  The book’s central message is thatorder and chaos coexist as part of a greater whole and both are to be embracedand understood.

I was introduced to the notion of phase space (page 35),which is a central idea in describing the behavior of many non-linearequations.  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 usefulnotion, but it is a concept that I can understand in the moment, put eludes mewhen I encounter it later on. Discussions of fixed point attractors, limit cycle attractors, andsaddles are much more understandable than others I encountered.  Interesting examples are used todescribe each—pike and trout (predator and prey).  Introducing fisherman and adding another dimension made itmuch more difficult to understand. Discussions of the torus as an attractor were elusive for me—kinda madesense and then I got lost.

This book had an entire chapter on strange attractors, whichI had hoped would help me break the code and understand them, but the conceptstill escapes me.  A useful analogyof water in a stream whose flow is interrupted by a rock was very helpful ingeneral in understanding the stages of behavior.  The message I took from this is that the amount of energy ina system is a major factor of whether or not the system will encounterchaos.   The notion ofsubdividing also poked its head up, but that is also proving to be an elusivenotion for me.

All the books that I read, including this, have introducedme to a great many mathematicians who have worked in areas related toturbulence.  This has been of morethan passing interest to me.  Veryinteresting to see how many minds can be working on issues and contribute to anoverall understanding of an issue. Leonardo drew pictures of turbulence, by subdividing vortexes, Landauand Hopf followed on.

The notion of fractals keeps appearing and it too iselusive.  On page 51, the metaphorof crumpling a piece of paper is used. “The more tightly it’s compressed, the more chaotic are its folds, andthe closer the two-dimensional surface moves to becoming a three-dimensionalsolid.  The Benard convection islike the crumpling paper, or a science fantasy character unable to choosebetween worlds.  In a desperate“effort” to escape to a higher dimension or return to a lower one, the currentwanders in the infinite byways of “indecision” between the two dimensions andthus crumples up.  The dimensionthat this “indecision” inhabits is therefore not a whole dimension (nottwo-dimensional or three-dimensional) but a fractional dimension.  And the shape the indecision traces isa strange attractor.”  [This is dense writing.   One could chew on this paragraph for a long time.]

A very interesting chapter used a formula for birthrateswith a limiter formula to show that a point of stability was reached for a widerange of birthrates.  It alsoshowed that for some birthrates there was a bifurcation and even multiplebifurcations which put the system into chaos.  He mentioned normalization, which I had encountered in aconsulting project in which some database experts kept mentioningnormalizing.  It was nice to seenormalization explained, however briefly.

The notion of intermittency was introduced and related to adiagram on page 31 which was extremely rich in detail and would be suitable forhours and hours of analysis.  Anexample of system generated static in a system was used and the notion thatmassive  networks were prone tointermittent issues was interesting. 

In this book and others the iterating of a formula was mostinteresting.  Issues such asperiodicity were introduced.  Onpage 72, a computer simulation of Poincare’s picture being folded and after 10iterations, it was a series of black and white lines, after 48 iterations therewere several images of Poincare and after 241 iterations, the original pictureof Poincare appeared.  The impactof rounding errors was explored and was shown to have great impact oniterations—leading to chaos by accumulation of rounding errors.  An example of a universe-sizedcomputer, rounding to 31 places would produce chaos in an iterative exampleafter only 100 iterations.  “At thespeed which normal computers do iterations, predictability vanishes withina fraction of a secondwhen highly nonlinear equations are concerned.”  [Emphasis is mine.]

In Chapter Four, the story of John Russell, who came acrossa solition—a solid wave that moved for several miles without dissipating istold.  This is an example of chaosholding things together.  The waveformed from the coupling of many smaller waves.  The speed of solitions are dependent on their height—a tallthin wave could catch up with a short fat one.  An interesting idea was put forward—the sound of a distantcannon will always be heard before theorder to fire.  This is because thesound of the cannon travels as a solition.  It’s interesting that Russell spent his entire life studyingthis phenomenon.  Tsunamis aresolitions.

David Ruelle’s Change and Chaos is a remarkably well written book, and at thebeginning appears to be quite approachable, but soon gets into areas that werebeyond my ability to grasp.  Ithink that the authors of all the books that I read were so immersed in theirsubject for so many years and for so much of their working life that they justcan not break down the subject any other way.  It’s such a large field of so much complexity, that it canonly be described to a certain level of simplicity.  After that, one needs a good sized “tool kit” of knowledgeand insight.  Ruelle is French andhas a beautiful command of the language. He also has a Puckish sense of humor.

Some examples:

From Acknowledgements: “Arthur Wightman and Laura Kang Ward fought nobly in defense of theEnglish language.”

From the Preface: “Speaking of scientific colleagues, some of them will be upset by myunglorious descriptions of scientists and the world of research.  For this I offer no apology:  if science is the research of truth, shouldone not also be truthful about how science is made.”

From Chapter One: “…we shall try to understand something of the triangular relationbetween the strangeness of mathematics, the strangeness of the physical world,and the strangeness of our own human mind.”

This book is as much a work of philosophy as it is one ofmath.

Does God Play Dice?  was an approachable bookinitially.  It defines chaos as“Stochastic behavior occurring in a deterministic system.”  Further, “…the definition is aparadox.  Deterministic behavior isruled by exact and unbreakable law. Stochastic behavior is the opposite:  lawless and irregular, governed by chance.  So chaos is ‘lawless behaviour governedentirely by law.’ “

Stewart mentions calculator chaos.  Using a calculator and iterating, some interesting patternsare seen.  Iterating x squared fora value less than one, yields a limit of zero within 9 iterations.  Using .54321 and pressing cosine buttonin the radian mode about 40 times, the number .739085133 appears and does notchange for successive iterations. Both are examples of convergence.

He iterated a tangent function 300,000 times and it neverconverged nor turned periodic, although it increased very slowly at times—by.0000001 per iteration.  This is anexample of intermittency, which is a typical behavior.

The square root button converges on 1.  The 1/x button eventually alternatesbetween .54321 and 1.840908673—an example of periodicity.  Iterating the equation of 2x squared-1with values of .054321 and .054322 looks very similar for awhile, but after the50^th iteration, it is completely different.

An example of kx squared – 1 shows that for k=1.4, there isa complicated cycling amongst 16 different values.  Chaos sets in at about k = 1.5 and gets worse as k isincreased.   But with chaos fullydeveloped 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 systemand tells of a Greek machine with gears that emulated the solar system, usingthe notion of epicycles.  He tellsof the evolution of thinking about the solar system.

Under the topic, The Reformulation Period, he uses a niceturn of phrase:  “  In 1750 Lagrange took up Euler’s ideasand produced from them an elegant and far-reaching reformulation ofdynamics.  Two important ideascrystallized out of his work.  Bothhad been around, as half-baked ideas, fordecades, but Lagrange baked them golden brown, took them out of theoven, and placed them on the bakery counter for all to admire, buy and consume.” [Emphasis is mine.] Hamilton refined this further to arrive at a single quantity—theHamiltonian which defines the total energy in terms of positions and momentums.

On pages 54 and 55, he mentions the state of scientificthinking at the beginning of the 20^th Century.  “As the 20^th Centuryunrolled, statistical methodology took its place alongside deterministicmodeling as an equal partner.  Anew word was coined to reflect the realization that even chance has its laws:  stochastic.  (The Greek word stochastikos means ‘skillful inaiming’…..  No longer was ordersynonymous 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, eachapplying only within its own sphere of influence.  … The two paradigms were equal partners….  Equal.  But different. Totally, irreconcilably different. Scientists knew they weredifferent, and they new why….  Betweensimplicity and complexity there could be no common ground.  But what one generation of scientistknows, beyond anyshadow of doubt, with a knowledge that is built into the very fabric of theirworld, is precisely what the succeeding generation will challenge and overturn.   If you know something that strongly, you don’t questionit.  If you don’t question it,you’re living by faith, not by science.”  [Bold and underlined emphasis is mine.]

This idea of being “trapped” by one’s knowledge is becoming acentral idea of what I’m picking up in this course.  Extending it out beyond math is interesting.  This morning I read a page one articleabout Condeleeza Rice in the New York Times.  It pointed out that her knowledge was based on the theory ofsuper-power relationships and that she was not equipped to consider a statelessthreat such as terrorism as being significant.

In describing Poincare, he says, “The voice of a man whotouched chaos…   And washorrified by it.”  He also statesthat Poincare “was perhaps the last mathematician able to roam at willthroughout every nook and cranny of his subject.”  (btw—a description of the foldable picture of Poincare isalso shown.)    “He gazedinto the abyss of chaos, he discerned some of the forms that lurked within; butthe abyss was still dark and he mistook for monstrosities some of the mostbeautiful things in mathematics. Poincare had the depth, but he lacked the means of illuminati.  It took another age, armed withPoincare’s own qualitative theory of differential equations, together withcomputers and other technical assistance, to shine some light into the chaoticdepths and reveal that beauty.  Butthey could never done it if Poincare hadn’t pioneered the way to the abyss’sedge.”

Poincare developed the field of topology, also characterizedas ‘rubber sheet geometry.’  Thiswas very mind stretching for me. Transformations from squares to circles and back and to a topologist allshapes are one. (quoted).  Poincarehad the notion of a plane intersecting a topological space—called his surfaceof section.  In a very involvedway, he could use this to show periodicity.

A quote on non-linear equations.  “Classical mathematics concentrated on linear equations fora sound pragmatic reason:  itcouldn’t solve anything else.  Incomparison to the unruly hooligan antics of a typical differential equation,linear ones are a bunch of choirboys. … So docile are linear equations that the classical mathematicians werewilling to compromise their physics to get them.  ….  So ingrained became the linear habitthat by the 1940s and 1950s many scientists and engineers knew littleelse.  ‘God would not be so unkind,‘ said a prominent engineer, ‘as to make the equations of nature nonlinear.’“  [Emphasis mine.]

“Really the whole language in which the discussion isconducted is topsy-turvy.  To calla general differential equation ‘nonlinear’ is rather like calling zoology‘nonpachydermology’.  But you see,we live in a world which for centuries acted as if the only animal in existencewas the elephant, which assumed that the holes in the skirting board must bemade by tiny elephants, which saw the soaring eagle as a wing-eared Dumbo, thetiger as an elephant with a rather short trunk and stripes, and whosetaxonomists resorted to corrective surgery so that the museum’s zoologicalcollection consisted entirely of lumbering grey pachyderms.  So ‘nonlinear’ it is.”

Pages 95 to 125 are devoted to the chapter entitled “StrangeAttractors.”  I was really stokedto see this, as I knew that this was a key to an understanding of chaos.  However, it reached beyond my abilityto understand and digest in the short period of this project.  There is an interesting diagram anddiscussion of sinks, saddles, limit cycles and sources which “typically” arethe 4 conditions met in a system of differential equations.  Next is a discussion of quasiperiodicmotions that take place on a torus, tied together by the idea of swinging a catin space.  Then a discussion ofSmale’s ideas of attractors, and Poincare’s sections and tenfold wrapping.  My head hurt lots here.

And finally ending up with “Cantor Cheese.”  This is a problem that starts with oneline and removes the middle third, resulting in two lines with a space equal totheir length in between them.  Nowiterate until you have LOTS of fragments. Points such as ½, 1/9, 2/9,7/9, and 8/9 escape removal.  (quote)  Anyway, I have seenmany versions of the Cantor set, but really don’t “get” what it’s going after.

This book went on to more difficult topics, but I gave upafter about 125 pages.

I was particularly disappointed in The Essence ofChaos by the “Father of Chaos”, EdwardLorenz.  I read 75 pages of whichmost was a discussion of cross slope differences, using a skiing analogy withmoguls.  I think there might bemore of interest later in the book, but I did not enjoy the writing style.

I scanned Newton’s Clock and was not up to reading more about chaos,particularly when it was focused on the solar system, which is a topic that haseluded me for sometime and for which I had no further energy.

Symmetry in Chaos is a visually stunning book. I didn’t 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 tome.  Tiling, wallpapers, fractalimages, geometric, and on and on. One interesting thing was the number of iterations of some of theseimages—30 million iterations of an equation were required to generateimages.  The colors were developedby the numbers of “hits” on a pixel during all the iterations.  Like I say, I really don’t understandit, but nonetheless, I am quite impressed with the results.

Web Sites

I am “cutting and pasting” from these sites, for info thatfills 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 atechnical 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 ordinarystudent, although it supposedly is aimed at undergraduate science majors.

Çambel (1993) defines chaoticsystems as "[s]ystems that upon analysis are found to be nonlinear,nonequilibrium, deterministic, dynamic, and that incorporate randomness so thatthey are sensitive to initial conditions, and have strange attractors".

http://www-chaos.umd.edu/

This a pretty random source of information and there is nooverall plan to the presentation of the story of Chaos.  The following quote was interestingnonetheless. 

From Poincare:

“…But it is not always so; it may happen that small differencesin the initial conditions produce very great ones in the final phenomena. A small error in the former will produce anenormous error in the latter. Prediction becomes impossible, and we have thefortuitous phenomenon. - in a 1903 essay "Science and Method" “

http://www.zeuscat.com/andrew/chaos/chaos.html

This site is about 8 years old and no longer addedto.  It did have some interesting“factoids”, which are included, as well as some droll humor at the end.

The LorenzianWaterwheel

Most casual armchair scientists have no access to uniformlysmooth boxes and elemental gases, much less instruments to measure therotational speed of a moving cylinder of gas.

A metaphor for the gaseous system is found in the Lorenzian waterwheel. This isa thought experiment. Imagine a waterwheel, with an arbitrary number ofbuckets, usually more than seven, spaced equally around its rim. The bucketsare mounted on swivels, much like Ferris-wheel seats, so that the buckets willalways open upwards. At the bottom of each bucket is a small hole. The entirewaterwheel 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 willtrickle into the top bucket, and immediately trickle out through the hole inthe bottom. Nothing happens. Real boring. Increase the flow just a bit,however, and the waterwheel will begin to revolve as the buckets fill up fasterthan they can empty. The heavier buckets containing more water let water out asthey descend, and when the water is gone, the now-light buckets ascend on theother side, ultimately to be refilled. The system is in a steady state; thewheel 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 boxesor heated gases, exhibits chaotic motion. Increase the flow of water, and strangethings will happen. The waterwheel will revolve in one direction as before, andthen suddenly jerk about and revolve in the other direction. The conditions ofthe buckets filling and emptying will no longer be so synchronous as tofacilitate just simple rotation; chaos has taken over.

What's a Fractal?

The Sierpinski Triangle raises all sorts of little questions that relate totopics in chaos theory not covered in the last few pages. For example, theSierpinski 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 atdifferent scales. This is rather dry. More clearly, a fractal shape will lookalmost, 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 SierpinskiTriangle. The first step in the geometric construction of the SierpinskiTriangle involved splitting a triangle up into three other triangles. When welook at the finished Sierpinski Triangle, we can zoom in on any of these threesub-triangles, and it will look exactly like the entire Sierpinski Triangleitself. 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 tobegin with a (usually, but not necessarily, equilateral) triangle (first figurebelow). Connect the midpoints of each side to form four separate triangles, andcut out the triangle in the center (second figure). For each of the threeremaining triangles, perform this same act (third figure). Iterate infinitely(final figure).

Theresult, as you can see, is the Sierpinski triangle. The geometric constructionof the Sierpinski triangle is the most intuitive way to generate thisfascinating fractal; however, it is only the tip of the Sierpinski iceberg.

As a total aside, I have found that methodically drawingthe Sierpinski triangle during boring lectures greatly relieves stress. If moreboring lectures are anticipated, draw a huge one, like one that spans an entiresheet of regular paper drawn to painstaking detail. After a few lectures yourboredom 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, reallyimpressively detailed (and large)Sierpinski Triangle which people will be really impressed with. They will saythings like "Man, that's cool!" and "Whoa, how'dja dothat!" and "Man, you must have been really bored."

http://www.duke.edu/~mjd/chaos/chaosh.html

This is an undergraduatepaper.

In addition to the famousSierpenski 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. Atthe middle of each side, add a new triangle one-third the size; and repeat thisprocess for an infinite amount of iterations. The length of the boundary is 3 X4/3 X 4/3 X 4/3...-infinity. However, the area remains less than the area of acircle drawn around the original triangle. What this means is that aninfinitely long line surrounds a finite area. The end construction of a KochSnowflake resembles the coastline of a shore.

On February 3rd, 1893, GastonMaurice Julia was born in Sidi Bel Abbes, Algeria. Julia was injured whilefighting in World War I and was forced to wear a leather strap across his facefor the rest of his life in order to protect and cover his injury. he spent alarge majority of his life in hospitals; therefore, a lot of his mathematicalresearch took place in the hospital. At the age of 25, Julia published a 199 pagemasterpiece entitled "Memoire sur l'iteration des fonctions." Thepaper dealt with the iteration of a rational function. With the publication ofthis paper came his claim to fame. Julia spent his life studying the iterationof polynomials and rational functions.

The Cantor set is simply thedust of points that remain. The number of these points are infinite, but theirtotal length is zero. Mandelbrot saw the Cantor set as a model for theoccurrence of errors in an electronic transmission line. Engineers saw periodsof errorless transmission, mixed with periods when errors would come in gusts.When these gusts of errors were analyzed, it was determined that they containederror-free periods within them. As the transmissions were analyzed to smallerand 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 theimplications are still ‘out there.’

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 andFractals are interesting, but not easy.

An excellent bibliography with lots of links to other websites.

http://mathforum.org/library/topics/chaos/

This site is fromDrexel 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 createfractal images.  It also has a Predator-PreyModel and a Game of Life that both looked interesting.

Chaos isapparently unpredictable behavior arising in a deterministic system because ofgreat sensitivity to initial conditions. Chaos arises in a dynamical system iftwo arbitrarily close starting points diverge exponentially, so that theirfuture behavior is eventually unpredictable.
What this means is that chaotic behavior, although appearing random, arisesfrom a very rigid cause. It also is highly sensitive to any disturbances, becauseevery change in the system will compound with time. Also, because of theextreme disorder, predicting the future path of the system is practicallyimpossible.

http://library.thinkquest.org/3120/

This site was put up by a group of high schoolstudents.  It is a testimony to howmuch 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 “nice” interactive explanation with virtuallyno 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 humanside of how knowledge about the world progresses along its uneven course.  The common theory that it builds on thepast is not as “true” as a picture that has many discontinuities, many of whichare discovered or developed by a single genius.  This genius is not beholden to the past and has a keenlyquestioning mind.  Also a mind ofextraordinary capaciousness or focus along a given line of inquiry.  These new ideas are often met withresistance and/or ridicule by those whose fame and fortune is tied to previousideas.

I have increased the scope of my thinking considerablyduring the process of writing this paper. I am intrigued by the wholeness of things—the ability of nature (and ofmankind) to hold opposites simultaneously.  It has been interesting to see how uncomfortable it has beenfor mankind to think of the world as having a chaotic element to it and howearnestly so many have worked to find order.  Sometimes the attempt has been to exert order where it doesnot exist.  This seems to have itsanalog in psychology—our “dark” sides are seen as disorders and not part of ourtotal personalities and other realms of human endeavor that seek to winnow outthe bad/imperfect and leave only the “good” behind.

The insight about non-linear being similar to non-pachydermwas most useful to me.

I think that I am more at ease with my “unknowing” about thedetails of chaos.  I am now morewilling to be happy about having been exposed to so much and to know that thereis 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, itis more and more apparent what an excellent job of presenting the topic ofchaos 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?  TheMathematics of Chaos.  Cambridge, MA: 

Basil Blackwell Inc., 1989.

Ruelle, David. Chance and Chaos.  Princeton, N.J.:  Princeton University Press, 1991

Peterson, Ivars. Newton’s 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. “Chaos theory and its implications for curriculum and teaching.”  Education  Summer1995, v115 n4 pgs 541 to 547

References from the www

There were 1,660,000 references called up by Googleon the search “chaos theory” and 457,000 references on the search “chaos theorybasics.”   Here are some ofthe more interesting ones that I plan to use.

Rae, Gregory. Chaos Theory.  Self published http://www.imho.com/grae/chaos/index.htmlaccessed on 3/22/2004

The Math Forum at Drexel University (Philadelphia)   This site  provides linksto 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

          Thissite has a tutorial on Chaos.  Thefive parts are:  The Philosophy ofDeterminism, Initial Conditions, Uncertainty of Measurements, Dynamical,Instabilities and Manifestations of Chaos.  This work was done at the Ilya Prigogine Center for Studiesin Statistical Mechanics and Complex Systems.  Prigogine was a 1977 Nobel Laureate in Chemistry who diedlast 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 internationalcompetition where student teams engage in collaborative, project-based learningto create educational websites. The winning entries form the ThinkQuest online

Glenn Elert   This looks like a very usefulwebsite.  Not sure who Glenn Elertis, 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 thatseems to be extremely well written and useful http://www.duke.edu/~mjd/chaos/chaosh.htmlaccessed on 3/22/2

Ho, Andrew seniorsoftware engineer at Tellme Networks, Inc.Self published website http://www.zeuscat.com/andrew/chaos/chaos.html004accessed on 3/22/2004

University of Maryland  Website for itsmultidisciplinary work in Chaos Theory http://www-chaos.umd.eduaccessed on 3/22/2004

Reil, Torsten  University ofOxford,  Introduction to ChaoticSystems http://users.ox.ac.uk/~quee0818/chaos/chaos.htmlaccessed on 3/22/2004

Wikipedia, The Free Encyclopedia  http://en.wikipedia.org/wiki/Chaos_theoryaccessed on 3/24/20004

Otherresources noted but not used.  Fromlibrary catalog at Mission College.

Applied Chaos Theory—Combel, QA 172.SC45 at West Valley

Impact of Chaos on Science and Society—Q172.5 C45I45 atWest Valley

Introduction to Chaotic Dynamics, Dwaney, QA 614.D48 atWest Valley

Natures Chaos, TR 721.658 at Mission College

Nova Video at WVC AV Circulation Desk

Chaos—New Science.  Gleick, QA 172.5C45G54 at West Valley.  This appears to be a seminal work, butfairly complex.

Chaostheory tamed    Williams,  QA 172.5C45W55 atWest Valley

ChaoticDynamics, An Introduction, Baker and Golub,  QA862.P4B35 at Mission College

ChaoticDynamics of non-linear relationships, Rasband, Q172.SC45 at Mission College

Complexity:  Life on the edge of chaos  Lewin, Roger   B105.c473L48