One of Feigenbaum's best known contributions to the subject is a heuristic theory, based on renormalization group ideas borrowed from statistical mechanics, tha

One of Feigenbaum's best known contributions to the subject is a heuristic theory, based on renormalization group ideas borrowed from statistical mechanics, that explains the period doubling cascade and lets one calculate these constants. The first mathematically rigorous computer-assisted proof was due to Oscar Lanford. I think now there are proofs that are not computer assisted, though I'm not as familiar with these developments. The Wikipedia article linked above links to some of this information. The following article has a slightly more technical summary together with references to many of the original papers (including Feigenbaum's): http://www.scholarpedia.org/article/Period_doubling http://www.scholarpedia.org/article/Period_doubling For those of you with a bit more matheamtical background and access to technical books, I like the summary of Feigenbaum's renormalization picture in Guckenheimer & Holmes.