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.

Relationship between $x_{n+2}$ and $x_{n}$ when $a=2.7$. Before the period doubling bifurcation occurs. The orbit converges to the fixed point $x_{f2}$.

Relationship between $x_{n+2}$ and $x_{n}$ when $a=3$. The tangent slope at the fixed point $x_{f2}$. is exactly 1, and a period doubling bifurcation occurs.

Relationship between $x_{n+2}$ and $x_{n}$ when $a=3.3$. The fixed point $x_{f2}$ becomes unstable, splitting into a periodic-2 stable cycle.

When $r=3.0$, we have a single intersection, with slope exactly $+1$, indicating that it is about to undergo a period-doubling.

When $r=3.4$, there are three intersection points, with the middle one unstable, and the two others stable.

When $r=3.45$, there are three intersection points, with the middle one unstable, and the two others having slope exactly $+1$, indicating that it is about to undergo another period-doubling.

When $r\approx 3.56994567$, there are infinitely many intersections, and we have arrived at chaos via the period-doubling route.

Scaling limit

Approach to the scaling limit as $r$ approaches $r^{*}=3.5699\cdots$ from below.

At the point of chaos $r^{*}=3.5699\cdots$, 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, converging to a fractal.

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 $\delta =4.6692016\cdots$.

For the wrong values of scaling factor $\alpha$, the map does not converge to a limit, but when $\alpha =2.5029\dots$, it converges.

At the point of chaos $r^{*}=3.5699\cdots$, as we repeat the functional equation iteration $f(x)\mapsto -\alpha f(f(-x/\alpha ))$ with $\alpha =2.5029\dots$, we find that the map does converge to a limit.

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.

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 ${\textstyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}$ is also the same function. This is an example of universality.

Logistic map approaching the period-doubling chaos scaling limit $r^{*}=3.84943\dots$ from below. At the limit, this has the same shape as that of $r^{*}=3.5699\cdots$, since all period-doubling routes to chaos are the same (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, ${\textstyle 3\delta \approx 2\alpha ^{2}}$, 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

$$g(x)=\lim _{n\rightarrow \infty }{\frac {1}{F^{\left(2^{n}\right)}(0)}}F^{\left(2^{n}\right)}\left(xF^{\left(2^{n}\right)}(0)\right)$$

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.

See also

• Logistic map
• Presentation function

Bibliography

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