Feigenbaum function

In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:[1] In the logistic map, we have a function

, and we want to study what happens when we iterate the map many times.

When the map falls into a stable fixed cycle of length

points, and the slope of the graph of

, we have a single intersection, with slope bounded in

, indicating that it is a stable single fixed point.

, the intersection point splits to two, which is a period doubling.

, there are three intersection points, with the middle one unstable, and the two others stable.

, the period doublings become infinite, and the map becomes chaotic.

Looking at the images, one can notice that at the point of chaos

, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees.

Further, as the period-doubling intervals become shorter and shorter, the ratio between two period-doubling intervals converges to a limit, the first Feigenbaum constant

For the wrong values, the map does not converge to a limit, but when it is

, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.

, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ...

This is an example of universality.We can also consider period-tripling route to chaos by picking a sequence of

is the lowest value in the period-

window of the bifurcation diagram.

converges to the fixed point to

In general, each period-multiplying route to chaos has its own pair of Feigenbaum constants.

, and the relation becomes exact as both numbers increase to infinity:

This functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade.

Discovered by Mitchell Feigenbaum and Predrag Cvitanović,[3] the equation is the mathematical expression of the universality of period doubling.

It specifies a function g and a parameter α by the relation with the initial conditions

For a particular form of solution with a quadratic dependence of the solution near x = 0, α = 2.5029... is one of the Feigenbaum constants.

The Feigenbaum function can be derived by a renormalization argument.

The Feigenbaum scaling function provides a complete description of the attractor of the logistic map at the end of the period-doubling cascade.

The attractor is a Cantor set, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size dn.

For a fixed dn the set of segments forms a cover Δn of the attractor.

The ratio of segments from two consecutive covers, Δn and Δn+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.

Approach to the scaling limit as approaches from below.
At the point of chaos , as we repeat the period-doublings , the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal.
For the wrong values of scaling factor , the map does not converge to a limit, but when , it converges.
At the point of chaos , as we repeat the functional equation iteration with , we find that the map does converge to a limit.
In the chaotic regime, , the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
Logistic map approaching the period-doubling chaos scaling limit from below. At the limit, this has the same shape as that of , since all period-doubling routes to chaos are the same (universality).