Feigenbaum's first constant

The first Feigenbaum constant 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.

It is given by the limit[1] where an are discrete values of a at the nth period doubling.

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:[3] 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.