In mathematics, specifically bifurcation theory, the Feigenbaum constants and are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum.
History
Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate. Feigenbaum made this discovery in 1975, and he officially published it in 1978.
The first constant
The first Feigenbaum constant or simply Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map
:<math>x_{i+1} = f(x_i),</math>
where is a function parameterized by the bifurcation parameter .
It is given by the limit:
:<math>\delta = \lim_{n\to\infty} \frac{a_{n-1} - a_{n-2{a_n - a_{n-1</math>
where are discrete values of at the th period doubling.
This gives its numerical value :
<math>\delta = 4.669\,201\,609\,102\,990\,671\,853\,203\,820\,466\ldots</math>
- A simple rational approximation is , which is correct to 5 significant values (when rounding). For more precision use , which is correct to 7 significant values.
- It is approximately equal to , with an error of 0.0047 %.
Illustration
Non-linear maps
To see how this number arises, consider the real one-parameter map
:<math>f(x) = a-x^2.</math>
Here is the bifurcation parameter, is the variable. The values of for which the period doubles (e.g. the largest value for with no orbit, or the largest with no orbit), are , etc. These are tabulated below:
:{| class="wikitable"
|-
!
! Period
! Bifurcation parameter ()
! Ratio
|-
| 1
|| 2
|| 0.75
|| —
|-
| 2
|| 4
|| 1.25
|| —
|-
| 3
|| 8
||
|| 4.2337
|-
| 4
|| 16
||
|| 4.5515
|-
| 5
|| 32
||
|| 4.6458
|-
| 6
|| 64
||
|| 4.6639
|-
| 7
|| 128
||
|| 4.6682
|-
| 8
|| 256
||
|| 4.6689
|-
|}
The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the logistic map
:<math>f(x) = ax(1-x)</math>
with real parameter and variable . Tabulating the bifurcation values again:
:{| class="wikitable"
|-
!
! Period
! Bifurcation parameter ()
! Ratio
|-
| 1
|| 2
|| 3
|| —
|-
| 2
|| 4
||
|| —
|-
| 3
|| 8
||
|| 4.7514
|-
| 4
|| 16
||
|| 4.6562
|-
| 5
|| 32
||
|| 4.6683
|-
| 6
|| 64
||
|| 4.6686
|-
| 7
|| 128
||
|| 4.6680
|-
| 8
|| 256
||
|| 4.6768
|-
|}
Fractals
right|thumb|201px|[[Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative- direction. The display center pans from (−1, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 to approximate the Feigenbaum ratio.]]
In the case of the Mandelbrot set for complex quadratic polynomial
:<math>f(z) = z^2 + c</math>
the Feigenbaum constant is the limiting ratio between the diameters of successive circles on the real axis in the complex plane (see animation ).
:{| class="wikitable"
|-
!
! Period =
! Bifurcation parameter ()
! Ratio <math>= \dfrac{c_{n-1} - c_{n-2{c_n - c_{n-1</math>
|-
| 1
|| 2
||
|| —
|-
| 2
|| 4
||
|| —
|-
| 3
|| 8
||
|| 4.2337
|-
| 4
|| 16
||
|| 4.5515
|-
| 5
|| 32
||
|| 4.6459
|-
| 6
|| 64
||
|| 4.6639
|-
| 7
|| 128
||
|| 4.6668
|-
| 8
|| 256
||
|| 4.6740
|-
|9
||512
||
||4.6596
|-
|10
||1024
||
||4.6750
|-
|...
||...
||...
||...
|-
|
||
|| ...
||
|}
Bifurcation parameter is a root point of period- component. This series converges to the Feigenbaum point = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.
thumb|right|[[Julia set for the Feigenbaum point]]
Other maps also reproduce this ratio; in this sense the Feigenbaum constant in bifurcation theory is analogous to Pi (number)| in geometry and in calculus.
The second constant
The second Feigenbaum constant or Feigenbaum reduction parameter
These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth). In fact, there is no known proof that either constant is even irrational.
The first proof of the universality of the Feigenbaum constants was carried out by Oscar Lanford—with computer-assistance—in 1982 (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987). Over the years, non-numerical methods were discovered for different parts of the proof, aiding Mikhail Lyubich in producing the first complete non-numerical proof.
Other values
The period-3 window in the logistic map also has a period-doubling route to chaos, reaching chaos at <math>r = 3.854 077 963 591\dots</math>, and it has its own two Feigenbaum constants: <math>\delta = 55.26, \alpha = 9.277</math>.
See also
- Bifurcation diagram
- Bifurcation theory
- Cascading failure
- Feigenbaum function
- List of chaotic maps
Notes
References
External links
- Feigenbaum Constant – from Wolfram MathWorld
:
:
- Feigenbaum constant – PlanetMath
- Pdf.