In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:[1]

Idea

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Period-doubling route to chaos

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In the logistic map,

  (1)

we have a function  , 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  , we would find that the graph of   and the graph of   intersects at   points, and the slope of the graph of   is bounded in   at those intersections.

For example, when  , we have a single intersection, with slope bounded in  , indicating that it is a stable single fixed point.

As   increases to beyond  , the intersection point splits to two, which is a period doubling. For example, when  , there are three intersection points, with the middle one unstable, and the two others stable.

As   approaches  , another period-doubling occurs in the same way. The period-doublings occur more and more frequently, until at a certain  , the period doublings become infinite, and the map becomes chaotic. This is the period-doubling route to chaos.

Relationship between   and   when  . Before the period doubling bifurcation occurs. The orbit converges to the fixed point  .
Relationship between   and   when  . The tangent slope at the fixed point  . is exactly 1, and a period doubling bifurcation occurs.
Relationship between   and   when  . The fixed point   becomes unstable, splitting into a periodic-2 stable cycle.
When  , we have a single intersection, with slope exactly  , indicating that it is about to undergo a period-doubling.
When  , there are three intersection points, with the middle one unstable, and the two others stable.
When  , there are three intersection points, with the middle one unstable, and the two others having slope exactly  , indicating that it is about to undergo another period-doubling.
When  , there are infinitely many intersections, and we have arrived at chaos via the period-doubling route.

Scaling limit

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Approach to the scaling limit as   approaches   from below.
 
At the point of chaos  , as we repeat the period-doublings , 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  , the curve of   looks like a fractal. Furthermore, as we repeat the period-doublings , 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   for a certain constant  :  then at the limit, we would end up with a function   that satisfies  . 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  .

For the wrong values of scaling factor  , the map does not converge to a limit, but when  , it converges.
 
At the point of chaos  , as we repeat the functional equation iteration   with  , we find that the map does converge to a limit.

The constant   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  , it converges. This is the second Feigenbaum constant.

Chaotic regime

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In the chaotic regime,  , the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.

In the chaotic regime,  , the limit of the iterates of the map, becomes chaotic dark bands interspersed with non-chaotic bright bands.

Other scaling limits

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When   approaches  , we have another period-doubling approach to chaos, but this time with periods 3, 6, 12, ... This again has the same Feigenbaum constants  . The limit of   is also the same function. This is an example of universality.

Logistic map approaching the period-doubling chaos scaling limit   from below. At the limit, this has the same shape as that of  , 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   such that   is the lowest value in the period-  window of the bifurcation diagram. For example, we have  , with the limit  . This has a different pair of Feigenbaum constants  .[2] And  converges to the fixed point to 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   such that   is the lowest value in the period-  window of the bifurcation diagram. Then we have  , with the limit  . This has a different pair of Feigenbaum constants  .

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,  , and the relation becomes exact as both numbers increase to infinity:  .

Feigenbaum-Cvitanović functional equation

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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,[3] 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 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   is approximately[4] 

Renormalization

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The Feigenbaum function can be derived by a renormalization argument.[5]

The Feigenbaum function satisfies[6]  for any map on the real line   at the onset of chaos.

Scaling function

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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.

See also

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Notes

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  1. ^ Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976
  2. ^ a b 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.
  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. 6 (3): 427–434. doi:10.1090/S0273-0979-1982-15008-X. ISSN 0273-0979.
  5. ^ 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)
  6. ^ Weisstein, Eric W. "Feigenbaum Function". mathworld.wolfram.com. Retrieved 2023-05-07.

Bibliography

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