Feigenbaum function

Feigenbaum functions are iterating nonlinear functions discovered by the mathematician Mitchell Feigenbaum.

A Feigenbaum function is in itself not particularly exciting; only when it is compounded in series, to produce a Feigenbaum sequence, does it produce interesting results. Nonlinear iterating functions feed the output of a function's previous iteration into the next iteration as that iteration's starting point. When an iterating function emits a stable pattern of output, it has stabilized. Depending on the mathematical operation the function performs, and the numerical values of any constants in the equation, an iterating function can stabilize on one value or some pattern of values. It can also never stabilize. Nonlinear iterating functions are a component of chaos math because a regular pattern of activity can, in the case of a non-stabilizing sequence, produce completely irregular results.

Certain Feigenbaum functions generate the Feigenbaum fractal, a recursively bifurcated fractal which starts as a single line and divides repeatedly, until it resembles a spiderweb and trails into dust.

To create a Feigenbaum sequence, start with a Feigenbaum function, such as:

x = x^2 - a

("x equals x to the second power, minus a.")

a is the constant, and x is the value of the previous iteration, or some small starting value. (x is often somewhere between 1 and 0.) Running this function repeatedly produces Feigenbaum sequences.

The Feigenbaum fractal has numerous structural parallels to the Mandelbrot set, mainly because the Mandelbrot set is essentially built of Feigenbaum functions. The formula for the Mandelbrot set is:

x = x^2 + a

Mitch Feigenbaum initially developed his fractal using only pen, paper, and a Hewlett Packard calculator, in the 1970s.