# 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 s_{i}
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:

(s_{i+1} - s_{i})/(s_{i+2} - s_{i+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.