Английская Википедия:Feigenbaum function

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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]

  • the solution to the Feigenbaum-Cvitanović functional equation; and
  • the scaling function that described the covers of the attractor of the logistic map

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,[2] the equation is the mathematical expression of the universality of period doubling. It specifies a function g and a parameter Шаблон:Mvar by the relation

<math> g(x) = - \alpha g( g(-x/\alpha ) ) </math>

with the initial conditions<math display="block">\begin{cases} 

   g(0) = 1, \\
   g'(0) = 0, \\
   g(0) < 0.

\end{cases}</math>For a particular form of solution with a quadratic dependence of the solution near Шаблон:Math is one of the Feigenbaum constants.

The power series of <math>g</math> is approximately[3]<math display="block">g(x) = 1 - 1.52763 x^2 + 0.104815 x^4 + 0.026705 x^6 + O(x^{8})</math>

Renormalization

The Feigenbaum function can be derived by a renormalization argument.[4]

The Feigenbaum function satisfies[5]<math display="block">g(x)=\lim _{n \rightarrow \infty} \frac{1}{F^{\left(2^n\right)}(0)} F^{\left(2^n\right)}\left(x F^{\left(2^n\right)}(0)\right)</math>for any map on the real line <math>F</math> 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 Δn+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.

See also

Notes

Шаблон:Reflist

Bibliography

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976
  2. ↑ Footnote on p. 46 of Feigenbaum (1978) states "This exact equation was discovered by P. Cvitanović during discussion and in collaboration with the author."
  3. ↑ Шаблон:Cite journal
  4. ↑ Шаблон:Cite book
  5. ↑ Шаблон:Cite web
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