# 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:
* the solution to the Feigenbaum-Cvitanović functional equation; and
* the scaling function that described the covers of the attractor of the logistic map

## Idea
### Period-doubling route to chaos

In the logistic map,
| $x_{n+1}=rx_{n}(1-x_{n}),$ | (1) |
we have a function $f_{r}(x)=rx(1-x)$, 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 $n$, we would find that the graph of $f_{r}^{n}$ and the graph of $x\mapsto x$ intersects at $n$ points, and the slope of the graph of $f_{r}^{n}$ is bounded in $(-1,+1)$ at those intersections.
For example, when $r=3.0$, we have a single intersection, with slope bounded in $(-1,+1)$, indicating that it is a stable single fixed point.
As $r$ increases to beyond $r=3.0$, the intersection point splits to two, which is a period doubling. For example, when $r=3.4$, there are three intersection points, with the middle one unstable, and the two others stable.
As $r$ approaches $r=3.45$, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain $r\approx 3.56994567$, the period doublings become infinite, and the map becomes chaotic. This is the period-doubling route to chaos.
$x_{n+2}$
$x_{n}$
$a=2.7$
$x_{f2}$
$x_{n+2}$
$x_{n}$
$a=3$
$x_{f2}$
$x_{n+2}$
$x_{n}$
$a=3.3$
$x_{f2}$
$r=3.0$
$+1$
$r=3.4$
$r=3.45$
$+1$
$r\approx 3.56994567$

### Scaling limit

Looking at the images, one can notice that at the point of chaos $r^{*}=3.5699\cdots$, the curve of $f_{r^{*}}^{\infty }$ looks like a fractal. Furthermore, as we repeat the period-doublings$f_{r^{*}}^{1},f_{r^{*}}^{2},f_{r^{*}}^{4},f_{r^{*}}^{8},f_{r^{*}}^{16},\dots$, 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 $\alpha$ for a certain constant $\alpha$:$f(x)\mapsto -\alpha f(f(-x/\alpha ))$ then at the limit, we would end up with a function $g$ that satisfies $g(x)=-\alpha g(g(-x/\alpha ))$. 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 $\delta =4.6692016\cdots$.
The constant $\alpha$ 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 $\alpha =2.5029\dots$, it converges. This is the second Feigenbaum constant.

### Chaotic regime

In the chaotic regime, $f_{r}^{\infty }$, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.

### Other scaling limits

When $r$ approaches $r\approx 3.8494344$, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants $\delta ,\alpha$. The limit of f(x)↦−αf(f(−x/α)){\textstyle f(x)\mapsto -\alpha f(f(-x/\alpha ))} 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 $r_{1},r_{2},\dots$ such that $r_{n}$ is the lowest value in the period-$3^{n}$ window of the bifurcation diagram. For example, we have $r_{1}=3.8284,r_{2}=3.85361,\dots$, with the limit $r_{\infty }=3.854077963\dots$. This has a different pair of Feigenbaum constants $\delta =55.26\dots ,\alpha =9.277\dots$. And $f_{r}^{\infty }$converges to the fixed point to$f(x)\mapsto -\alpha f(f(f(-x/\alpha )))$As 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 $r_{1},r_{2},\dots$ such that $r_{n}$ is the lowest value in the period-$4^{n}$ window of the bifurcation diagram. Then we have $r_{1}=3.960102,r_{2}=3.9615554,\dots$, with the limit $r_{\infty }=3.96155658717\dots$. This has a different pair of Feigenbaum constants $\delta =981.6\dots ,\alpha =38.82\dots$.
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, 3δ≈2α$^2${\textstyle 3\delta \approx 2\alpha ^{2}}, and the relation becomes exact as both numbers increase to infinity: $\lim \delta /\alpha ^{2}=2/3$.

## 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 α by the relation
    $g(x)=-\alpha g(g(-x/\alpha ))$
with the initial conditions${\begin{cases}g(0)=1,\\g'(0)=0,\\g''(0)<0.\end{cases}}$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 power series of $g$ is approximately$g(x)=1-1.52763x^{2}+0.104815x^{4}+0.026705x^{6}+O(x^{8})$

## Renormalization

The Feigenbaum function can be derived by a renormalization argument.
The Feigenbaum function satisfies$g(x)=\lim _{n\to \infty }{\frac {1}{F^{\left(2^{n}\right)}(0)}}F^{\left(2^{n}\right)}\left(xF^{\left(2^{n}\right)}(0)\right)$ for any map on the real line $F$ 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 d$_n$. For a fixed d$_n$ the set of segments forms a cover Δ$_n$ of the attractor. The ratio of segments from two consecutive covers, Δ$_n$ and Δ$_{n+1}$ can be arranged to approximate a function σ, the Feigenbaum scaling function.