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,

$x_{n+1}=r{x}_n(1-x_n),$ (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.

File:ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 2.7）.png

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 $x_{f2}$.

File:ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 3）.png

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.

File:ロジスティック写像2回反復グラフの周期倍化分岐の様（a = 3.3）.png

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.

File:Logistic map iterates, r=3.0.svg

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

File:Logistic iterates 3.4.svg

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

File:Logistic iterates r=3.45.svg

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.

File:Logistic iterates with r=3.56994567.svg

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

Scaling limit

File:Logistic map approaching the scaling limit.webm

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

File:Logistic iterates, together, r=3.56994567.svg

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 $f_{r^{*}}^{\mathrm{\infty }}$ 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

$\delta =4.6692016\cdots$

.

File:Logistic scaling with varying scaling factor.webm

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

File:Logistic scaling limit, r=3.56994567.svg

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, $f_r^{\mathrm{\infty }}$, the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.

File:Logistic map in the chaotic regime.webm

In the chaotic regime, $f_r^{\mathrm{\infty }}$, 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

$f(x)\mapsto -\alpha f(f(-x/\alpha ))$

is also the same function. This is an example of universality.

File:Logistic map approaching the period-3 scaling limit.webm

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 $r_{\mathrm{\infty }}=3.854077963\dots$. This has a different pair of Feigenbaum constants $\delta =55.26\dots ,\alpha =9.277\dots$. And $f_r^{\mathrm{\infty }}$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 $r_{\mathrm{\infty }}=3.96155658717\dots$. 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 α by the relation

$g(x)=-\alpha g(g(-x/\alpha ))$

with the initial conditions$$\{\begin{array}{l}g(0)=1,\\ g^{\prime }(0)=0,\\ g^{″}(0)<0.\end{array}$$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$$g(x)=\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{F^{\left(2^n\right)}(0)}{F}^{\left(2^n\right)}(x{F}^{\left(2^n\right)}(0))$$ 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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