In the logistic map,
we have a function f r ( x ) = r x ( 1 − x ) {\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 n {\displaystyle n} , we would find that the graph of f r n {\displaystyle f_{r}^{n}} and the graph of x ↦ x {\displaystyle x\mapsto x} intersects at n {\displaystyle n} points, and the slope of the graph of f r n {\displaystyle f_{r}^{n}} is bounded in ( − 1 , + 1 ) {\displaystyle (-1,+1)} at those intersections.
For example, when r = 3.0 {\displaystyle r=3.0} , we have a single intersection, with slope bounded in ( − 1 , + 1 ) {\displaystyle (-1,+1)} , indicating that it is a stable single fixed point.
As r {\displaystyle r} increases to beyond r = 3.0 {\displaystyle r=3.0} , the intersection point splits to two, which is a period doubling. For example, when r = 3.4 {\displaystyle r=3.4} , there are three intersection points, with the middle one unstable, and the two others stable.
As r {\displaystyle r} approaches r = 3.45 {\displaystyle r=3.45} , another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain r ≈ 3.56994567 {\displaystyle r\approx 3.56994567} , the period doublings become infinite, and the map becomes chaotic. This is the period-doubling route to chaos.
Looking at the images, one can notice that at the point of chaos r ∗ = 3.5699 ⋯ {\displaystyle r^{*}=3.5699\cdots } , the curve of f r ∗ ∞ {\displaystyle 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 , … {\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 α {\displaystyle \alpha } for a certain constant α {\displaystyle \alpha } : f ( x ) ↦ − α f ( f ( − x / α ) ) {\displaystyle f(x)\mapsto -\alpha f(f(-x/\alpha ))} then at the limit, we would end up with a function g {\displaystyle g} that satisfies g ( x ) = − α g ( g ( − x / α ) ) {\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 δ = 4.6692016 ⋯ {\displaystyle \delta =4.6692016\cdots } .
The constant α {\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 α = 2.5029 … {\displaystyle \alpha =2.5029\dots } , it converges. This is the second Feigenbaum constant.
In the chaotic regime, f r ∞ {\displaystyle f_{r}^{\infty }} , the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.
When r {\displaystyle r} approaches r ≈ 3.8494344 {\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 δ , α {\displaystyle \delta ,\alpha } . The limit of f ( x ) ↦ − α f ( f ( − x / α ) ) {\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 r 1 , r 2 , … {\displaystyle r_{1},r_{2},\dots } such that r n {\displaystyle r_{n}} is the lowest value in the period- 3 n {\displaystyle 3^{n}} window of the bifurcation diagram. For example, we have r 1 = 3.8284 , r 2 = 3.85361 , … {\displaystyle r_{1}=3.8284,r_{2}=3.85361,\dots } , with the limit r ∞ = 3.854077963 … {\displaystyle r_{\infty }=3.854077963\dots } . This has a different pair of Feigenbaum constants δ = 55.26 … , α = 9.277 … {\displaystyle \delta =55.26\dots ,\alpha =9.277\dots } .2 And f r ∞ {\displaystyle f_{r}^{\infty }} converges to the fixed point to f ( x ) ↦ − α f ( f ( f ( − x / α ) ) ) {\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 r 1 , r 2 , … {\displaystyle r_{1},r_{2},\dots } such that r n {\displaystyle r_{n}} is the lowest value in the period- 4 n {\displaystyle 4^{n}} window of the bifurcation diagram. Then we have r 1 = 3.960102 , r 2 = 3.9615554 , … {\displaystyle r_{1}=3.960102,r_{2}=3.9615554,\dots } , with the limit r ∞ = 3.96155658717 … {\displaystyle r_{\infty }=3.96155658717\dots } . This has a different pair of Feigenbaum constants δ = 981.6 … , α = 38.82 … {\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.3
Generally, 3 δ ≈ 2 α 2 {\textstyle 3\delta \approx 2\alpha ^{2}} , and the relation becomes exact as both numbers increase to infinity: lim δ / α 2 = 2 / 3 {\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 by Mitchell Feigenbaum and Predrag Cvitanović,4 the equation is the mathematical expression of the universality of period doubling. It specifies a function g and a parameter α by the relation
with the initial conditions { g ( 0 ) = 1 , g ′ ( 0 ) = 0 , g ″ ( 0 ) < 0. {\displaystyle {\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 {\displaystyle g} is approximately5 g ( x ) = 1 − 1.52763 x 2 + 0.104815 x 4 + 0.026705 x 6 + O ( x 8 ) {\displaystyle g(x)=1-1.52763x^{2}+0.104815x^{4}+0.026705x^{6}+O(x^{8})}
The Feigenbaum function can be derived by a renormalization argument.6
The Feigenbaum function satisfies7 g ( x ) = lim n → ∞ 1 F ( 2 n ) ( 0 ) F ( 2 n ) ( x F ( 2 n ) ( 0 ) ) {\displaystyle g(x)=\lim _{n\to \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 {\displaystyle F} at the onset of chaos.
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 Δn+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.
Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976 http://chaosbook.org/extras/mjf/LA-6816-PR.pdf ↩
Delbourgo, R.; Hart, W.; Kenny, B. G. (1985-01-01). "Dependence of universal constants upon multiplication period in nonlinear maps". Physical Review A. 31 (1): 514–516. Bibcode:1985PhRvA..31..514D. doi:10.1103/PhysRevA.31.514. ISSN 0556-2791. https://link.aps.org/doi/10.1103/PhysRevA.31.514 ↩
Footnote on p. 46 of Feigenbaum (1978) states "This exact equation was discovered by P. Cvitanović during discussion and in collaboration with the author." ↩
Iii, Oscar E. Lanford (May 1982). "A computer-assisted proof of the Feigenbaum conjectures". Bulletin (New Series) of the American Mathematical Society. 6 (3): 427–434. doi:10.1090/S0273-0979-1982-15008-X. ISSN 0273-0979. https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-6/issue-3/A-computer-assisted-proof-of-the-Feigenbaum-conjectures/bams/1183548786.full ↩
Feldman, David P. (2019). Chaos and dynamical systems. Princeton. ISBN 978-0-691-18939-0. OCLC 1103440222.{{cite book}}: CS1 maint: location missing publisher (link) 978-0-691-18939-0 ↩
Weisstein, Eric W. "Feigenbaum Function". mathworld.wolfram.com. Retrieved 2023-05-07. https://mathworld.wolfram.com/FeigenbaumFunction.html ↩