Several decades ago, Edward Lorenz noticed something seemingly minor: his computational weather models yielded very different results if he changed the input variables only a tiny bit. Lorenz published a paper on this topic in 1963 and that led to the modern field of chaos.

In *Physics Today*, Adilson Motter and David Campbell survey the field fifty years later and what we’ve learned about chaos and our limits to predictability. Here’s a description of the Lorenz system (and chaotic attractors in general):

A chaotic attractor is the example par excellence of a chaotic set. A chaotic set has uncountably many chaotic trajectories; on such a set, any point that lies in the neighborhood of a given point will also, with probability one, give rise to a chaotic trajectory. Yet no matter the proximity of those two points, in the region between them will lie points of infinitely many periodic orbits. In mathematical parlance, the periodic orbits constitute a countable, zero-measure, but dense set of points embedded in the chaotic set, analogous to the rational numbers embedded in the set of real numbers. Not only will trajectories that lie on the attractor behave chaotically, any point lying within the attractor’s basin of attraction will also give rise to chaotic trajectories that converge to the attractor.

Of course, chaos is found in many places far beyond weather models.

Written By: Samuel Arbesman

continue to source article at wired.com

How about a short and comprehensible definition of the term ‘chaos’ in an article about its 50th anniversary? I’m all for pretty pictures, but sadly, many mathematicians like to cover things by a smoke-screen.

If you read the article you would find a link paper to a free full text download from PhysicsToday.

Sorry for being so direct, but if you think mathematicians like blowing smoke, you either don’t know any mathematicians or you don’t know enough mathematics to go much beyond the surface. Sometimes the only accessible examples are so trivial it’s like “so what?” and then the next step up is a big one. Don’t blame mathematicians because a lot of the subject is often inaccessible if you don’t have the right background and hard to present in 25 words or less. It’s easier to write handwaving popular science books about cosmology without mathematics than about non-trivial mathematics without mathematics. Having said that, this link Chaos Theory is nevertheless guaranteed not to be a LMGTFY.

It covers a lot of ground, uses a lot of technical jargon, has an extensive bibliography (light/medium/heavy). This is where I started getting together the following information for you. Without knowing what level of mathematical sophistication a random reader has it’s hard to pick just the right reference. I tried to curate a selection of introductory material at various levels of technicality. You’ll have to start trying them on until you find something that fits your background and interest. (I hope you aren’t the only one who reads this!)

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.– E. LorenzChaos, Wolfram MathWorld (light/medium). I liked this because I think you will appreciate the comments about definitions, or lack thereof. It includes a more verbose reformulation of Lorenz (above):

So a simple, if slightly imprecise, way of describing chaos is “chaotic systems are distinguished by sensitive dependence on initial conditions and by having evolution through phase space that appears to be quite random.Chaos

by James P. Crutchfield, J. Doyne Farmer, Norman H. Packard, and Robert S. Shaw,

Scientific American, December, 1986 Vol. 254 No. 12, 46-57. (light/medium)Chaos Theory Simplified: Just Follow the Bouncing Droplet from Scientific American (light)

The Chaos Hypertextbook An introductory primer on chaos and fractals (medium)

Chaos, Encyclopedia of Mathematics (heavy)

High Anxieties – The Mathematics of Chaos, BBC Documentary (on Youtube in 9 parts) Part 1/9 (light)The Chaos Theory of Evolution looks interesting. (light)

No wonder most people prefer religion and the supernatural to science. They are

somuch easier to understand.In reply to #3 by joost:Exactly. Since there are no constraints on truth, all the creator of a religion has to consider is saleability.

We figured out the easy science first. This seduced us into thinking the universe is constrained to be comprehensible by 10% of homo sapiens.

Note how relativity and quantum mechanics boggles pretty well everyone’s intuition. Scientists have close their eyes and do the math, trusting that math has not failed them before.

There was a period in my own early life when the idea of Australians walking around upside down seemed absurd, even though both parents explained everything carefully using a globe. I felt there must be some mistake.

Mathematicians and physicists are always hoping for one-liner equations and one page proofs. Computers can manage gigabytes of detail. Perhaps there will be no way at all of intuiting what they discover.

When I was studying math back in the 60s, I noticed nearly all the great proofs had a trick, that for some reason academia had declared legit. I considered such proofs illegitimate. You must lay out what operations are permitted ahead of time otherwise you could be fooling yourself and on some level assuming what it is you are proving. I preferred “busy work” proofs that went on for pages and pages, but never did anything in the least clever.

Regular proofs seemed like so much handwaving, assuming things even more unlikely than the the thing we were trying to prove. We bothered to formalise just what sorts of deduction were legit.

My profs disagreed thoroughly. They hated my proofs. I had great hope that computers would force mathematicians to get completely explicit on what was legit in a proof. I wanted it to be a mechanical process to verify a proof, not an aesthetic one. I have some vindication. The computer generated proof for Fermat’s theorem was done in my plodding style.

In reply to #3 by joost:No, religion just makes being confused more tolerable.

There is a great essay by Dawkins in his book The Devil’s Chaplain that talks about pseudoscience and post modernism and mentions chaos theory. Not that he implies Chaos Theory is pseudoscience, not at all, but he points out how purveyors of pseudoscience (e.g. Depak Chopra, Robert Lanza) love to latch on to things like Chaos Theory and use it to give their BS a veneer of science. These days Quantum is the new Chaos Theory and in the essay Dawkins shows how postmodernist authors use such theories in ways that are almost interchangable. Essentially find something that sounds mysterious and that most people don’t understand and you can then use it to support whatever you want.

In reply to #1 by EtotheiPi:I just want to second what Whitefaven said. In my experience mathematicians don’t like to “cover things” at all in fact they love explaining stuff. Its just that they also like to be very precise and they care more about precision than about being easy to understand. I wish there were more people like that actually.

I’m sorry, I may not have expressed myself very well (english is just my third language). I guess, I was just frustrated because the author missed an opportunity to explain an important topic clearly. The fact that you can always find out what the core issue is by googling it or by following external links doesn’t help. Despite the legendary quarrel between physicists (like myself) and mathematicians, I admire their work tremendously. The sad thing is just that they sometimes don’t try to put themselves in their audiences shoes. I know that strange attractors are zero-measure sets, but it’s much more important that chaos theory describes deterministic dynamical systems which are extremely sensitive to changes in the initial conditions, or in Edward Lorentz’s words: “When the present determines the future, but the approximate present does not approximately determine the future.”

I wish more people were able to explain complicated topics as clearly as Richard Dawkins (of course without dumbing anything down!).

In reply to #8 by EtotheiPi:I agree. People who write about science should write as if their lives depended on whether the intended audience could understand them easily and properly.

In reply to #4 by Roedy:This doesn’t sound like it was written by someone who actually studied mathematics seriously. This could be a complete misimpression. I’d really be interested in learning more about what sort of mathematics you were studying at what level where you had these experiences. What little (next to nothing) I know about the proofs of the 4-Color Problem and Fermat’s Theorem is that after decades and mountains of mind-boggling theoretical work, computational methods were used to brute force large numbers of special cases.

The closest I can come to what you’re talking about are issues about existence proofs, proof by contradiction, use of the axiom of choice. You reminded me of a visiting professor I met as an undergrad in the ’70s who was in Constructive Analysis. Constructive Mathematics is a serious program so you must have been engaged in some real fights over the philosophy of mathematics.

In reply to #9 by joost:I disagree because you are putting it all on the communicator and freeing the audience of any responsibility. Communication is important but I feel no performance anxiety or responsibility if my cat fails to understand what I’m saying to you.

I propose that people who pass through K-12 should study mathematics and science as if their lives depended on it so when they are the audience for information with any scientific or numerical content they they will more easily understand and think about it. No one should get past ’12’ without having learned learned enough about probability and statistics to know that state lotteries are not fair games and why. (oops, we all know why that’ll never happen!)

In reply to #11 by whiteraven:I’m sorry, but only medical doctors get that privilege (you can never blame them for the state of

yourhealth). Educators these days must seduce, beg, make house calls and even whip themselves until their audience understands. It’s just the way it is.In reply to #8 by EtotheiPi:I’ll agree to this: One of the many places where chaos is found is in the self-referntial paragraph introduced by “Here’s a description of the Lorenz system (and chaotic attractors in general)”. The reason I included the link to Wolfram MathWorld is that practically half the article is a complaint about getting a definition of chaos.