Feigenbaum Constant

The Feigenbaum constant arises in Chaos Theory when studying the behavior of certain types of functions as they undergo period-doubling bifurcations. It is named after the physicist Mitchell Feigenbaum, who discovered it in the 1970s.

First example was found by Feigenbaum studying the Logistic Map, where period double bifurcations happen as the parameter r is increased. However it was later found to apply to all one dimensional maps with single quadratic maxima. Meaning, every chaotic system corresponds to this description will bifurcate at the same rate.

Roughly defined as:

$$\delta =\underset{n\to \infty }{\lim }\frac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}}=\mathrm{4.6692016091029906718532038204662016172581855774757686327456513435...}$$

Not to be confused with the Feigenbound Constant, which is a different mathematical constant related to the bounds of certain functions.

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Second Constant

There are actuall two Feigenbaum constants. The second one $\alpha$, also known as the Feigenbaum reduction parameter, is defined as:

$$\alpha =\underset{n\to \infty }{\lim }\frac{d_n}{d_{n+1}}=\mathrm{2.50290787509589282228390287321821578...}$$

It is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold).