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."