Feigenbaum function

Idea

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-doublingsf^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 .[[File:Logistic scaling with varying scaling factor.webm|thumb|480x480px|For the wrong values of scaling factor \alpha , the map does not converge to a limit, but when \alpha = 2.5029\dots , it converges.]]

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.[[File:Logistic_map_approaching_the_period-3_scaling_limit.webm|thumb|482x482px|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 = 3.854 077 963\dots. This has a different pair of Feigenbaum constants \delta= 55.26\dots, \alpha = 9.277\dots. And f^\infty_rconverges to the fixed point tof(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 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) \end{cases}For a particular form of solution with a quadratic dependence of the solution near is one of the Feigenbaum constants.

The power series of g is approximatelyg(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 satisfiesg(x) = \lim_{n\to\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 Δn of the attractor. The ratio of segments from two consecutive covers, Δn and *Δ*n+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.

Notes

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References

1. [http://chaosbook.org/extras/mjf/LA-6816-PR.pdf Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976]
2. (1985-01-01). "Dependence of universal constants upon multiplication period in nonlinear maps". Physical Review A.
3. Footnote on p. 46 of Feigenbaum (1978) states "This exact equation was discovered by P. Cvitanović during discussion and in collaboration with the author."
4. Iii, Oscar E. Lanford. (May 1982). "A computer-assisted proof of the Feigenbaum conjectures". Bulletin (New Series) of the American Mathematical Society.
5. Feldman, David P.. (2019). "Chaos and dynamical systems".
6. Weisstein, Eric W.. "Feigenbaum Function".