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

Period-doubling route to chaos
In the logistic map, 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.

Scaling limit
Looking at the images, one can notice that at the point of chaos $$r^* = 3.5699\cdots$$, the curve of $$f^{\infty}_{r^*}$$ looks like a fractal. Furthermore, as we repeat the period-doublings$$f^{1}_{r^*}, f^{2}_{r^*}, f^{4}_{r^*}, f^{8}_{r^*}, f^{16}_{r^*}, \dots$$, 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, $$f^\infty_r$$, 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) \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 $$r_\infty = 3.854 077 963\dots$$. This has a different pair of Feigenbaum constants $$\delta= 55.26\dots, \alpha = 9.277\dots$$. And $$f^\infty_r$$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, $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 $$ by the relation
 * $$ 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 $x = 0, &alpha; = 2.5029...$ is one of the Feigenbaum constants.

The power series of $$g$$ is approximately $$g(x) = 1 - 1.52763 x^2 + 0.104815 x^4 + 0.026705 x^6 + O(x^{8})$$

Renormalization
The Feigenbaum function can be derived by a renormalization argument.

The Feigenbaum function satisfies $$g(x)=\lim _{n \rightarrow \infty} \frac{1}{F^{\left(2^n\right)}(0)} F^{\left(2^n\right)}\left(x F^{\left(2^n\right)}(0)\right)$$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 dn. For a fixed dn the set of segments forms a cover &Delta;n of the attractor. The ratio of segments from two consecutive covers, &Delta;n and &Delta;n+1 can be arranged to approximate a function &sigma;, the Feigenbaum scaling function.