4.669...............

Feigenbaum constants [show article only]hover over links in text for more info

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List of numbers Irrational and suspected irrational numbers γ ζ(3) √2 √3 √5 φ ρ δS e π δ

Feigenbaum constant δ expresses the limit of the ratio of distances between consecutive bifurcation diagram on L_i / L_i + 1

In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the mathematician Mitchell Feigenbaum.

Contents

• 1 History
• 2 The first constant
  • 2.1 Illustration
    • 2.1.1 Non-linear maps
    • 2.1.2 Fractals
• 3 The second constant
• 4 Properties
• 5 See also
• 6 Notes
• 7 References
• 8 External links

History[edit]

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, it was discovered in 1978.^[1]

The first constant[edit]

The first Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map

${\displaystyle x_{i+1}=f(x_{i}),}$

where f(x) is a function parameterized by the bifurcation parameter a.
It is given by the limit^[2]

${\displaystyle \delta =\lim _{n\to \infty }{\frac {a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}}=4.669\,201\,609\,\ldots ,}$

where a_n are discrete values of a at the n-th period doubling.
Here is this number to 30 decimal places (sequence A006890 in the OEIS): δ = 4.669201609102990671853203821578…

Illustration[edit]

Non-linear maps[edit]

To see how this number arises, consider the real one-parameter map

${\displaystyle f(x)=a-x^{2}.}$

Here a is the bifurcation parameter, x is the variable, the values of a for which the period doubles (e.g. the largest value for a with no period-2 orbit, or the largest a with no period-4 orbit), are a₁, a₂ etc. These are tabulated below:^[3]

n	Period	Bifurcation parameter (an)	Ratio an−1 − an−2/an − an−1
1	2	0.75	—
2	4	1.25	—
3	8	7000136809889999999♠1.3680989	4.2337
4	16	7000139404620000000♠1.3940462	4.5515
5	32	7000139963120000000♠1.3996312	4.6458
6	64	7000140082860000000♠1.4008286	4.6639
7	128	7000140108530000000♠1.4010853	4.6682
8	256	7000140114020000000♠1.4011402	4.6689