# 费根鲍姆常数

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

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

Feigenbaum 常数 δ 表示 l < sub > i /l < sub > i + 1 上连续分枝图间距离比的极限值

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 physicist Mitchell J. Feigenbaum.

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 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. It was discovered in 1975.[1][2]

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 1975.

Feigenbaum 最初在 logistic 映射中把第一个常数与倍周期分岔联系起来，但也证明了它对所有一维映射都具有单一二次最大值。由于这种普遍性，每一个符合这种描述的混沌系统都将以相同的速率分叉。它于1975年被发现。

## The first constant

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

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

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

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

< math > x _ { i + 1} = f (xi) ，</math >

where f(x) is a function parameterized by the bifurcation parameter a.

where is a function parameterized by the bifurcation parameter .

It is given by the limit[3]

It is given by the limit

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

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

4.669,201,609，ldots，</math >

where an are discrete values of a at the n-th period doubling.

where are discrete values of at the -th period doubling.

### Names

• Feigenbaum bifurcation velocity
• delta

### Value

• A simple rational approximation is 4 * 307 / 263

### Illustration

#### Non-linear maps

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

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

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

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

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:[4]

Here is the bifurcation parameter, is the variable. The values of for which the period doubles (e.g. the largest value for with no period-2 orbit, or the largest with no period-4 orbit), are , etc. These are tabulated below:

{ | class = “ wikitable”
n Period Period 句号 Bifurcation parameter (an) Bifurcation parameter () 分岔参数() Ratio 模板:Sfrac a − a}}}} a − a}}}}
1 1 1 2 2 2 0.75 0.75 0.75
2 2 2 4 4 4 1.25 1.25 1.25
3 3 3 8 8 8 模板:Val 4.2337 4.2337 4.2337
4 4 4 16 16 16 模板:Val 4.5515 4.5515 4.5515
5 5 5 32 32 32 模板:Val 4.6458 4.6458 4.6458
6 6 6 64 64 64 模板:Val 4.6639 4.6639 4.6639
7 7 7 128 128 128 模板:Val 4.6682 4.6682 4.6682
8 8 8 256 256 256 模板:Val 4.6689 4.6689 4.6689

|}

The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the logistic map

The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the logistic map

$\displaystyle{ f(x) = a x (1 - x) }$

$\displaystyle{ f(x) = a x (1 - x) }$

= a x (1-x) </math >

with real parameter a and variable x. Tabulating the bifurcation values again:[5]

with real parameter and variable . Tabulating the bifurcation values again:

{ | class = “ wikitable”
n Period Period 句号 Bifurcation parameter (an) Bifurcation parameter () 分岔参数() Ratio 模板:Sfrac a − a}}}} a − a}}}}
1 1 1 2 2 2 3 3 3
2 2 2 4 4 4 模板:Val
3 3 3 8 8 8 模板:Val 4.7514 4.7514 4.7514
4 4 4 16 16 16 模板:Val 4.6562 4.6562 4.6562
5 5 5 32 32 32 模板:Val 4.6683 4.6683 4.6683
6 6 6 64 64 64 模板:Val 4.6686 4.6686 4.6686
7 7 7 128 128 128 模板:Val 4.6692 4.6692 4.6692
8 8 8 256 256 256 模板:Val 4.6694 4.6694 4.6694

|}

#### Fractals

Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-x 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.

[[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.]]

[在 Mandelbrot 集合中的自相似性表现为放大一个圆形特征，同时平移反方向。显示中心平移从(- 1,0)到(- 1.31,0) ，而视图放大从0.5 × 0.5到0.12 × 0.12，接近 Feigenbaum 比率。]

In the case of the Mandelbrot set for complex quadratic polynomial

In the case of the Mandelbrot set for complex quadratic polynomial

$\displaystyle{ f(z) = z^2 + c }$

$\displaystyle{ f(z) = z^2 + c }$

= z ^ 2 + c </math >

the Feigenbaum constant is the ratio between the diameters of successive circles on the real axis in the complex plane (see animation on the right).

the Feigenbaum constant is the ratio between the diameters of successive circles on the real axis in the complex plane (see animation on the right).

Feigenbaum 常数是复平面上实轴上连续圆的直径之比(见右图)。

{ | class = “ wikitable”
n Period = 2n Period = 句号 = Bifurcation parameter (cn) Bifurcation parameter () 分岔参数() Ratio $\displaystyle{ = \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} }$ Ratio $\displaystyle{ = \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} }$ 比率 < math > = dfrac { c _ { n-1}-c _ { n-2}{ c _ n-c _ { n-1}}} </math >
1 1 1 2 2 2 模板:Val
2 2 2 4 4 4 模板:Val
3 3 3 8 8 8 模板:Val 4.2337 4.2337 4.2337
4 4 4 16 16 16 模板:Val 4.5515 4.5515 4.5515
5 5 5 32 32 32 模板:Val 4.6458 4.6458 4.6458
6 6 6 64 64 64 模板:Val 4.6639 4.6639 4.6639
7 7 7 128 128 128 模板:Val 4.6682 4.6682 4.6682
8 8 8 256 256 256 模板:Val 4.6689 4.6689 4.6689
9 9 9 512 512 512 模板:Val
10 10 10 1024 1024 1024 模板:Val

|}

Bifurcation parameter is a root point of period-2n component. This series converges to the Feigenbaum point c = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant.

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.

Other maps also reproduce this ratio, in this sense the Feigenbaum constant in bifurcation theory is analogous to π in geometry and e in calculus.

Other maps also reproduce this ratio, in this sense the Feigenbaum constant in bifurcation theory is analogous to Pi (number)| in geometry and e (mathematical constant)| in calculus.

## The second constant

The second Feigenbaum constant 模板:OEIS,

The second Feigenbaum constant ,

α = 模板:Gaps,
 = ,

 = ,


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 α when the ratio between the lower subtine and the width of the tine is measured.[6]

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 when the ratio between the lower subtine and the width of the tine is measured.

These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).[6]

These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth). There is also no known proof that either constant is irrational.

A simple rational approximation is (13/11) * (17/11) * (37/27).

The first proof of the universality of the Feigenbaum constants carried out by Oscar Lanford in 1982 (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987) was computer-assisted. 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.

1982年，Oscar Lanford 首次证明了费根鲍姆常数的普遍性(1987年，日内瓦大学的让·彼埃尔·埃克曼和 Peter Wittwer 做了一个小小的修正) ，这是计算机辅助的。多年来，对于证明的不同部分发现了非数值方法，这有助于 Mikhail Lyubich 制作第一个完整的非数值证明。

## Properties

Both numbers are believed to be transcendental, although they have not been proven to be so.[7] There is also no known proof that either constant is irrational.

The first proof of the universality of the Feigenbaum constants carried out by Oscar Lanford in 1982[8] (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987[9]) was computer-assisted. 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.[10]

|first=Keith

|last=Briggs

| last = Briggs

|journal=Mathematics of Computation

| journal = 计算数学

|date=July 1991

|pages=435–439

| 页数 = 435-439

|volume=57

57

|title=A Precise Calculation of the Feigenbaum Constants

## Notes

|bibcode = 1991MaCom..57..435B |doi = 10.1090/S0025-5718-1991-1079009-6

|bibcode = 1991MaCom..57..435B |doi = 10.1090/S0025-5718-1991-1079009-6

1. Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976
2. Chaos: An Introduction to Dynamical Systems, K.T. Alligood, T.D. Sauer, J.A. Yorke, Springer, 1996,
3. Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th Edition), D. W. Jordan, P. Smith, Oxford University Press, 2007, .
4. Alligood, p. 503.
5. Alligood, p. 504.
6. Nonlinear Dynamics and Chaos, Steven H. Strogatz, Studies in Nonlinearity ,Perseus Books Publishing, 1994,
7. Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
8. Lanford III, Oscar (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Amer. Math. Soc. 6 (3): 427–434. doi:10.1090/S0273-0979-1982-15008-X.
9. Eckmann, J. P.; Wittwer, P. (1987). "A complete proof of the Feigenbaum conjectures". Journal of Statistical Physics. 46 (3–4): 455. Bibcode:1987JSP....46..455E. doi:10.1007/BF01013368.
10. Lyubich, Mikhail (1999). "Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness Conjecture". Annals of Mathematics. 149 (2): 319–420. arXiv:math/9903201. Bibcode:1999math......3201L. doi:10.2307/120968. JSTOR 120968.

|issue=195 }}

195}

## References

|first=Keith

• Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences Springer, 1996,

|last=Briggs

| last = Briggs

}}

|pages=435–439

|volume=57

|first1=David

1 = David

|title=A Precise Calculation of the Feigenbaum Constants

|bibcode = 1991MaCom..57..435B |doi = 10.1090/S0025-5718-1991-1079009-6

|issue=195 }}

|title= Feigenbaum constants to 1018 decimal places

| title = 费根鲍姆常数小数点后1018位

• (Thesis). 22 March 1999

}}

|last=Briggs

|publisher=University of Melbourne

|year=1997

|degree=PhD

|title=Feigenbaum scaling in discrete dynamical systems

}}

Category:Dynamical systems

Category:Mathematical constants

Category:Bifurcation theory