Numbers of Feigenbaum

In Mathematical, the numbers of constant Feigenbaum or of Feigenbaum are two Nombre S real discovered by the Mathématicien Mitchell Feigenbaum in 1975. Both express reports/ratios appearing in the diagrams of junction of the Théorie of chaos.

The diagrams of junction relate to the values Limite S taken by the continuations of the type $x\_\; \{n+1\}\; =\; \backslash \; driven\; F\; (x\_n)$ where F is a function real, definite positive and three times derivable on and having a single maximum on this interval (i.e. without relative maximum ) noted x_m . For a given function, in lower part of a certain value of μ , the continuation leads to a single limit. With the top of that value, but below an other, the continuation ends up oscillating between two values, then above an other value, to oscillate around four, etc the values of μ separating two intervals are called junctions and are noted μ₁ , μ₂ , etc

The first constant of Feigenbaum is defined like the limit of the relationship between two successive intervals of the junction:

$\backslash \; delta\; =\; \backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; frac\; \{\backslash \; mu\_\; \{n+1\}\; -\; \backslash \; mu\_n\}\; \{\backslash \; mu\_\; \{n+2\}\; -\; \backslash \; mu\_\; \{n+1\}\}$

Within the framework of the logistic Continuation where $x\_\; \{n+1\}\; =\; \backslash \; driven\; x\_n\; (1-x\_n)$ (initially studied by Feigenbaum):

$\backslash \; delta\; \backslash ,$ = 4,66920160910299067185320382…

It appears that it is also about the relationship between the diameters of two successive circles on the axis of the Ensemble of Mandelbrot. Consequently, any chaotic system which obeys this description will fork at the same speed. The first constant of Feigenbaum can be used to predict when the chaos will arrive in such system.

The second constant of Feigenbaum is defined like the limit of the relationship between two successive distances between the branches closest to x_m (the maximum of the function F ):

$\backslash \; alpha\; =\; \backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; frac\; \{d\_n\}\; \{d\_\; \{n+1\}\}$

Always within the framework of the logistic function:

$\backslash \; alpha\; \backslash ,$ = 2,502907875095892822283902873218…

These constants apply to a broad class of dynamic systems. An assumption is that these two numbers are transcendent although that was never still proven.

External bond

• Article on the constants of Feigenbaum on MathWorld.com

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