Chaotic Math…

"The mathematics of chaotic systems produces the same effect at every scale. Tell me how precise you want to be, and I can introduce my little germ of instability one decimal place further along; it may take a few more repetitions before the whole system's state becomes unpredictable, but the inevitability of chaos remains. The conventional image has the flap of a butterfly's wings in Brazil causing a storm in China, but even this is a needlessly gross impetus. The physicist David Ruelle, a major figure in chaos theory, gives a convincing demonstration that suspending the gravitational effect on our atmosphere of one electron at the limit of the observable universe would take no more than two weeks to make a difference in Earth's weather equivalent to having rain rather than sun during a romantic picnic."

-- From Michael and Ellen Kaplan's book "Chances Are…"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Meditating... on Chaos

Food for thought… an interesting passage from chapter 85 (on chaos theory, also known as "bifurcation theory") of Richard Elwes' "Math In 100 Key Breakthroughs" (pgs. 345-7):

"How can one produce a random number? In the late 1940's, John von Neumann proposed a very strange answer to that question.  He suggested that applying a simple algebraic rule a few times should do the job. The rule is to begin with some number, call it x, and then multiply x by (1 - x), and multiply the result by 4. That is to say: x --> 4 X x X (1 - x).
"There does not seem to be anything especially 'random' about this bit of algebra. Once the initial number is chosen, say x = 0.1, the result of applying the rule is then completely predetermined. But a little experimentation reveals von Neumann's insight. The sequence produced by this rule runs: 0.1, 0.36, 0.9216, 0.2890, 0.8219, 0.5854, 0.9708, and so on (each number given to 4 decimal places). There does not seem to be much of a pattern here, and in fact that is no illusion. You can extend the sequence for as long as you like and in fact no pattern will emerge. Someone who did not know the rule being used would find it virtually impossible to distinguish between this sequence and one produced by a genuinely random physical process such as radioactive decay."…

"Today, von Neumann's rule is known as the logistic map, and it is one of the simplest examples of mathematical chaos, a phenomenon which has been recognized in many different situations…."

"In von Neumann's pseudorandom number generator, everything rests on the number 4, known as the parameter. Changing that value completely alters the behavior of the system. If one replaces 4 with a new parameter of 2, the logistic map ceases to be chaotic. Instead, for any starting value, the sequence will quickly home in on a fixed value of 0.5. This is known as an attracting point of the system.
"Increase the parameter from 2 to 3.4, and something new occurs. After a while, the sequence will endlessly flicker back and forth between two values around 0.84 and 0.45. This is known as an attracting 2-cycle. Raise the parameter a little higher to 3.5, and this is replaced with an attracting 4-cycle, and then at 3.55, an attracting 8-cycle, and so on. As the parameter increases, the length of the attracting cycle keeps doubling 15, 32, 64, and so on. This behavior is what chaos theorists call a sequence of bifurcations."

He goes on to explain that the bifurcations end once the parameter hits a certain threshold value known as the Feigenbaum point (named after chaos theorist Mitchell Feigenbaum). Beyond that point (like "4" in the example) the produced sequence will act chaotically forever, producing the famous "butterfly effect" whereby two sequences beginning at only slightly different starting values "end up entirely unrecognizable from each other."

More On Complexity...

A great introduction (with several links) today to complexity theory and chaos, from RJ Lipton here:

http://rjlipton.wordpress.com/2011/02/21/butterflies-and-complexity-theories/

...and I learned that the "Butterfly Effect" actually goes all the way back to a Ray Bradbury novel... (since I don't read science fiction, or any fiction for that matter, I never knew that... or, did the notion come to Bradbury from yet someone or something else... who got it from.....?!)

Chaotic Chaos

Quote... unquote....

Been tied up with other things lately, but will put up for contemplation this passage from chapter 9 of Michael and Ellen Kaplan's book "Chances Are...," wherein they're discussing the work of Edward Lorenz, chaos theory, and deterministic systems:

"The mathematics of chaotic systems produces the same effect at every scale. Tell me how precise you want to be, and I can introduce my little germ of instability one decimal place further along; it may take a few more repetitions before the whole system's state becomes unpredictable, but the inevitability of chaos remains. The conventional image has the flap of a butterfly's wings in Brazil causing a storm in China, but even this is a needlessly gross impetus. The physicist David Ruelle, a major figure in chaos theory, gives a convincing demonstration that suspending the gravitational effect on our atmosphere of one electron at the limit of the observable universe would take no more than two weeks to make a difference in Earth's weather equivalent to having rain rather than sun during a romantic picnic."