monkey, it interests me! Though I'm only half-way through so far. The author seems to have given an excellent description of some of Feigenbaum's original insights -- E.g., sensitive dependence on initial conditions; simple, "self-replicating," periodically iterated rules.
I gather the simple rule is a mathematical equation into which numbers are "fed." Except I don't think the word "self-replicating" is quite accurate because, in Feigenbaum's original experiment, the next successive number generated by the equation refers back to the iteration immediately preceding it, and must take that number into account (i.e., by adding it to itself). Then comes the next iteration, with the new number "fed" into the equation, etc., etc. It's a "feedback loop." Ad infinitum.
This is fascinating stuff, monkey. Give me some time to digest it a bit. I may be back. best, bb.