Feigenbaum constants
Template:Short description Template:Use dmy dates Template:Infobox non-integer number
In mathematics, specifically bifurcation theory, the Feigenbaum constants Template:IPAc-en<ref>Template:Citation</ref> Template:Mvar and Template:Mvar 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.
HistoryEdit
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,<ref>Template:Cite journal</ref><ref>Template:Cite book</ref> and he officially published it in 1978.<ref>Template:Cite journal</ref>
The first constantEdit
The first Feigenbaum constant or simply Feigenbaum constant<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:Mvar 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 Template:Math is a function parameterized by the bifurcation parameter Template:Mvar.
It is given by the limit:<ref>Template:Cite book</ref>
- <math>\delta = \lim_{n\to\infty} \frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}}</math>
where Template:Mvar are discrete values of Template:Mvar at the Template:Mvarth period doubling.
This gives its numerical value (sequence A006890 in the OEIS):
<math>\delta = 4.669\,201\,609\,102\,990\,671\,853\,203\,820\,466\ldots</math>
- A simple rational approximation is Template:Sfrac, which is correct to 5 significant values (when rounding). For more precision use Template:Sfrac, which is correct to 7 significant values.
- It is approximately equal to Template:Math, with an error of 0.0047 %.
IllustrationEdit
Non-linear mapsEdit
To see how this number arises, consider the real one-parameter map
- <math>f(x) = a-x^2.</math>
Here Template:Mvar is the bifurcation parameter, Template:Mvar is the variable. The values of Template:Mvar for which the period doubles (e.g. the largest value for Template:Mvar with no Template:Nowrap orbit, or the largest Template:Mvar with no Template:Nowrap orbit), are Template:Math, Template:Math etc. These are tabulated below:<ref>Alligood, p. 503.</ref>
Template:Mvar Period Bifurcation parameter (Template:Mvar) Ratio Template:Math 1 2 0.75 — 2 4 1.25 — 3 8 Template:Val 4.2337 4 16 Template:Val 4.5515 5 32 Template:Val 4.6458 6 64 Template:Val 4.6639 7 128 Template:Val 4.6682 8 256 Template:Val 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 Template:Mvar and variable Template:Mvar. Tabulating the bifurcation values again:<ref>Alligood, p. 504.</ref>
Template:Mvar Period Bifurcation parameter (Template:Mvar) Ratio Template:Math 1 2 3 — 2 4 Template:Val — 3 8 Template:Val 4.7514 4 16 Template:Val 4.6562 5 32 Template:Val 4.6683 6 64 Template:Val 4.6686 7 128 Template:Val 4.6680 8 256 Template:Val 4.6768
FractalsEdit
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 on the right).
Template:Mvar Period = Template:Math Bifurcation parameter (Template:Mvar) Ratio <math>= \dfrac{c_{n-1} - c_{n-2}}{c_n - c_{n-1}}</math> 1 2 Template:Val — 2 4 Template:Val — 3 8 Template:Val 4.2337 4 16 Template:Val 4.5515 5 32 Template:Val 4.6459 6 64 Template:Val 4.6639 7 128 Template:Val 4.6668 8 256 Template:Val 4.6740 9 512 Template:Val 4.6596 10 1024 Template:Val 4.6750 ... ... ... ... Template:Math Template:Val...
Bifurcation parameter is a root point of period-Template:Math component. This series converges to the Feigenbaum point Template:Mvar = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.
Other maps also reproduce this ratio; in this sense the Feigenbaum constant in bifurcation theory is analogous to [[Pi (number)|Template:Pi]] in geometry and Template:Math in calculus.
The second constantEdit
The second Feigenbaum constant or Feigenbaum reduction parameter<ref name=":0" /> Template:Mvar is given by (sequence A006891 in the OEIS):
- <math>\alpha = 2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218\ldots</math>
It is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold). A negative sign is applied to Template:Mvar when the ratio between the lower subtine and the width of the tine is measured.<ref name="NonlinearDynamics">Template:Cite book</ref>
These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).<ref name="NonlinearDynamics" />
A simple rational approximation is Template:Sfrac × Template:Sfrac × Template:Sfrac = Template:Sfrac.
PropertiesEdit
Both numbers are believed to be transcendental, although they have not been proven to be so.<ref>Template:Cite thesis</ref> 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<ref>Template:Cite journal</ref> (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987<ref>Template:Cite journal </ref>). 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.<ref>Template:Cite journal</ref>
Other valuesEdit
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>.<ref>Template:Cite journal</ref><ref>Template:Cite book</ref>Template:Rp
See alsoEdit
NotesEdit
ReferencesEdit
- Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences Springer, 1996, Template:Isbn
- Template:Cite journal
- Template:Cite thesis
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External linksEdit
- Feigenbaum constant – PlanetMath
- Julia notebook for calculating Feigenbaum constant<ref>{{#invoke:citation/CS1|citation
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