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:

Idea

Period-doubling route to chaos

In the logistic map,
we have a function, and we want to study what happens when we iterate the map many times. The map might fall into a fixed point, a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length, we would find that the graph of and the graph of intersects at points, and the slope of the graph of is bounded in at those intersections.
For example, when, we have a single intersection, with slope bounded in, indicating that it is a stable single fixed point.
As increases to beyond, the intersection point splits to two, which is a period doubling. For example, when, there are three intersection points, with the middle one unstable, and the two others stable.
As approaches, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain, the period doublings become infinite, and the map becomes chaotic. This is the period-doubling route to chaos.

Scaling limit

Looking at the images, one can notice that at the point of chaos, the curve of looks like a fractal. Furthermore, 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.
This suggests to us a scaling limit: if we repeatedly double the function, then scale it up by for a certain constant : then at the limit, we would end up with a function that satisfies. 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.
The constant can be numerically found by trying many possible values. For the wrong values, the map does not converge to a limit, but when it is, it converges. This is the second Feigenbaum constant.

Chaotic regime

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

Other scaling limits

When approaches, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12,... This again has the same Feigenbaum constants. The limit of is also the same function. This is an example of universality.
We can also consider period-tripling route to chaos by picking a sequence of such that is the lowest value in the period- window of the bifurcation diagram. For example, we have, with the limit. This has a different pair of Feigenbaum constants. And converges to the fixed point toAs another example, period-4-pling has a pair of Feigenbaum constants distinct from that of period-doubling, even though period-4-pling is reached by two period-doublings. In detail, define such that is the lowest value in the period- window of the bifurcation diagram. Then we have, with the limit. This has a different pair of Feigenbaum constants.
In general, each period-multiplying route to chaos has its own pair of Feigenbaum constants. In fact, there are typically more than one. For example, for period-7-pling, there are at least 9 different pairs of Feigenbaum constants.
Generally,, and the relation becomes exact as both numbers increase to infinity:.

Feigenbaum-Cvitanović functional equation

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, the equation is the mathematical expression of the universality of period doubling. It specifies a function g and a parameter by the relation
with the initial conditionsFor a particular form of solution with a quadratic dependence of the solution
near is one of the Feigenbaum constants.
The power series of is approximately

Renormalization

The Feigenbaum function can be derived by a renormalization argument.
The Feigenbaum function satisfies for any map on the real line at the onset of chaos.

Scaling function

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.