In the study ofdynamical systems the termFeigenbaum function has been used to describe two differentfunctions introduced by the physicistMitchell Feigenbaum:^[1]

• the solution to the Feigenbaum-Cvitanovićfunctional equation; and
• the scaling function that described the covers of theattractor of thelogistic map

In the logistic map,

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

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 afixed 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 theperiod-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 atchaos via the period-doubling route.

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

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

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$f(x)\mapsto -\alpha f(f(-x/\alpha ))$ is also the same function. This is an example ofuniversality.

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${\displaystyle r_{\mathrm{\infty }}=3.854077963\dots }$. This has a different pair of Feigenbaum constants${\displaystyle \delta =55.26\dots ,\alpha =9.277\dots }$.^[2] And${\displaystyle f_r^{\mathrm{\infty }}}$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${\displaystyle r_{\mathrm{\infty }}=3.96155658717\dots }$. 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,$3\delta \approx 2{\alpha }^2$, and the relation becomes exact as both numbers increase to infinity:${\displaystyle lim\delta /{\alpha }^2=2/3}$.

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 byMitchell Feigenbaum andPredrag Cvitanović,^[3] the equation is the mathematical expression of theuniversality of period doubling. It specifies a functiong and a parameterα by the relation

${\displaystyle g(x)=-\alpha g(g(-x/\alpha ))}$

with the initial conditionsFor a particular form of solution with a quadratic dependence of the solutionnearx = 0,α = 2.5029... is one of theFeigenbaum constants.

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

The Feigenbaum function can be derived by arenormalization argument.^[5]

The Feigenbaum function satisfies^[6]$${\displaystyle g(x)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{F^{\left(2^n\right)}(0)}{F}^{\left(2^n\right)}\left(x{F}^{\left(2^n\right)}(0)\right)}$$ for any map on thereal line${\displaystyle F}$ at the onset of chaos.

The Feigenbaum scaling function provides a complete description of theattractor of thelogistic map at the end of the period-doubling cascade. The attractor is aCantor set, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal sized_n. For a fixedd_n the set of segments forms a coverΔ_n of the attractor. The ratio of segments from two consecutive covers,Δ_n andΔ_n+1 can be arranged to approximate a functionσ, the Feigenbaum scaling function.

• Logistic map
• Presentation function

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