Deterministic Chaos

Universality of the Logistic Map Bifurcation Diagram

To see the quantitative similarities in the period-doubling sequence of these diagrams, recall the scaling behavior discovered by Feigenbaum.

Feigenbaum found the period-doubling bifurcation parameters si using Newton's method on a programmable pocket calculator.

The speed at which Newton's method converges depends in part on how close the initial guess is to the true solution, and since Feigenbaum's pocket calculator was not especially fast, he began looking for patterns to produce better initial guesses.

He discovered the series seemed to be converging geometrically and computed the limit of the ratio of successive differences:

(si+1 - si)/(si+2 - si+1)

to try to predict the next value.

The limit of these ratios is the Feigenbaum delta constant,

delta = 4.6692016091029... .

Feigenbaum's model for understanding this process relied upon the quadratic nature of the logistic map, so he was discouraged by the Metropolis, Stein, and Stein result that many other functions exhibit the same qualitative bifurcation behavior.

Some time later he computed the limit of ratios for s*sin(pi*x) and obtained

4.6692016091029... .

This must be magic: to many decimal places, the ratio of differences of successive doubling parameters is independent of the particular function being iterated, at least for many functions. How can this be?

Return to Universality of the Logistic Map Bifurcation Diagram.