Period-doubling route to chaos

In the logistic map,

${\displaystyle x_{n+1}=rx_{n}(1-x_{n}),}$

(1)

we have a function ${\displaystyle f_{r}(x)=rx(1-x)}$, 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 ${\displaystyle n}$, we would find that the graph of ${\displaystyle f_{r}^{n}}$ and the graph of ${\displaystyle x\mapsto x}$ intersects at ${\displaystyle n}$ points, and the slope of the graph of ${\displaystyle f_{r}^{n}}$ is bounded in ${\displaystyle (-1,+1)}$ at those intersections.

For example, when ${\displaystyle r=3.0}$, we have a single intersection, with slope bounded in ${\displaystyle (-1,+1)}$, indicating that it is a stable single fixed point.

As ${\displaystyle r}$ increases to beyond ${\displaystyle r=3.0}$, the intersection point splits to two, which is a period doubling. For example, when ${\displaystyle r=3.4}$, there are three intersection points, with the middle one unstable, and the two others stable.

As ${\displaystyle r}$ approaches ${\displaystyle r=3.45}$, another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain ${\displaystyle r\approx 3.56994567}$, the period doublings become infinite, and the map becomes chaotic. This is the period-doubling route to chaos.

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

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

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

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

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

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

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

Scaling limit

Looking at the images, one can notice that at the point of chaos ${\displaystyle r^{*}=3.5699\cdots }$, the curve of ${\displaystyle f_{r^{*}}^{\infty }}$ looks like a fractal. Furthermore, as we repeat the period-doublings${\displaystyle f_{r^{*}}^{1},f_{r^{*}}^{2},f_{r^{*}}^{4},f_{r^{*}}^{8},f_{r^{*}}^{16},\dots }$, 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 ${\displaystyle \alpha }$ for a certain constant ${\displaystyle \alpha }$:

$${\displaystyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}$$

then at the limit, we would end up with a function ${\displaystyle g}$ that satisfies ${\displaystyle g(x)=-\alpha g(g(-x/\alpha ))}$. 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 ${\displaystyle \delta =4.6692016\cdots }$.

The constant ${\displaystyle \alpha }$ 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 ${\displaystyle \alpha =2.5029\dots }$, it converges. This is the second Feigenbaum constant.

Chaotic regime

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

Other scaling limits

When ${\displaystyle r}$ approaches ${\displaystyle r\approx 3.8494344}$, we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants ${\displaystyle \delta ,\alpha }$. The limit of ${\textstyle f(x)\mapsto -\alpha f(f(-x/\alpha ))}$ is also the same function. This is an example of universality.

We can also consider period-tripling route to chaos by picking a sequence of ${\displaystyle r_{1},r_{2},\dots }$ such that ${\displaystyle r_{n}}$ is the lowest value in the period-${\displaystyle 3^{n}}$ window of the bifurcation diagram. For example, we have ${\displaystyle r_{1}=3.8284,r_{2}=3.85361,\dots }$, with the limit ${\displaystyle r_{\infty }=3.854077963\dots }$. This has a different pair of Feigenbaum constants ${\displaystyle \delta =55.26\dots ,\alpha =9.277\dots }$.^[2] And ${\displaystyle f_{r}^{\infty }}$converges to the fixed point to

$${\displaystyle f(x)\mapsto -\alpha f(f(f(-x/\alpha )))}$$

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 ${\displaystyle r_{1},r_{2},\dots }$ such that ${\displaystyle r_{n}}$ is the lowest value in the period-${\displaystyle 4^{n}}$ window of the bifurcation diagram. Then we have ${\displaystyle r_{1}=3.960102,r_{2}=3.9615554,\dots }$, with the limit ${\displaystyle r_{\infty }=3.96155658717\dots }$. This has a different pair of Feigenbaum constants ${\displaystyle \delta =981.6\dots ,\alpha =38.82\dots }$.

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.^[2]

Generally, ${\textstyle 3\delta \approx 2\alpha ^{2}}$, and the relation becomes exact as both numbers increase to infinity: ${\displaystyle \lim \delta /\alpha ^{2}=2/3}$.