In mathematics, specifically bifurcation theory, the Feigenbaum constants /ˈfaɪɡənbaʊm/[1] δ and α are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map.

As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate.

[4] The first Feigenbaum constant or simply Feigenbaum constant[5] δ is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map where f (x) is a function parameterized by the bifurcation parameter a.

To see how this number arises, consider the real one-parameter map Here a is the bifurcation parameter, x is the variable.

The same number arises for the logistic map with real parameter a and variable x. Tabulating the bifurcation values again:[8] In the case of the Mandelbrot set for complex quadratic polynomial the Feigenbaum constant is the limiting ratio between the diameters of successive circles on the real axis in the complex plane (see animation on the right).

Other maps also reproduce this ratio; in this sense the Feigenbaum constant in bifurcation theory is analogous to π in geometry and e in calculus.

A negative sign is applied to α when the ratio between the lower subtine and the width of the tine is measured.

[9] These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).