费根鲍姆常数

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文件:Feigenbaum.png
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.

在数学中,特别是分岔理论,费根鲍姆常数是两个数学常数,它们都表示一个非线性映射的分枝图的比率。它们以物理学家米切尔 · 费根鲍姆的名字命名。


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

第一个 Feigenbaum 常数是单参数映射每个周期倍增之间每个分岔区间与下一个分岔区间的极限比

[math]\displaystyle{ x_{i+1} = f(x_i), }[/math]

[math]\displaystyle{ x_{i+1} = f(x_i), }[/math]

< 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 .

其中一个参数为分支参数的函数。


It is given by the limit[3]

It is given by the limit

它是由极限给出的

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

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

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.

其中第-周期的离散值加倍。


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

要了解这个数字是如何产生的,请考虑真正的单参数映射

[math]\displaystyle{ f(x)=a-x^2. }[/math]

[math]\displaystyle{ f(x)=a-x^2. }[/math]

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:

这里是分岔参数,是变量。周期为两倍的值(例如。无周期 -2轨道的最大值,或无周期 -4轨道的最大值,等等。这些数据表列如下:


{ | 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

最后一列的比值收敛于第一个 Feigenbaum 常数。对于后勤地图也会出现相同的数字

[math]\displaystyle{ f(x) = a x (1 - x) }[/math]

[math]\displaystyle{ f(x) = a x (1 - x) }[/math]

= 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:

带有真实参数和变量。再次列出分歧值:


{ | 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

文件:Mandelbrot zoom.gif
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

在 Mandelbrot 集合的情况下,对于复杂的二次多项式

[math]\displaystyle{ f(z) = z^2 + c }[/math]

[math]\displaystyle{ f(z) = z^2 + c }[/math]

= 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 [math]\displaystyle{ = \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} }[/math] Ratio [math]\displaystyle{ = \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} }[/math] 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
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.

分岔参数是周期分量的根点。这个序列收敛于 Feigenbaum 点 =-1.401155..。最后一列的比值收敛于第一个 Feigenbaum 常数。


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.

其他地图也重现了这个比率,在这个意义上,分岔理论的 Feigenbaum 常数类似于几何学中的 Pi (数) | 和微积分学中的 e (数学常数) | 。


The second constant

The second Feigenbaum constant 模板:OEIS,

The second Feigenbaum constant ,

第二个 Feigenbaum 常数,

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.

是一个齿的宽度与其两个亚齿之一的宽度之比(除了最靠近折叠的齿)。当测量下齿与齿宽之比时,应用负号。


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.

这些数字适用于一大类动态系统(例如,水龙头滴水到人口增长)。也没有已知的证据证明这两个常数都是无理的。


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]


See also

|first=Keith

第一名: Keith

|last=Briggs

| last = Briggs

|url=http://www.ams.org/journals/mcom/1991-57-195/S0025-5718-1991-1079009-6/S0025-5718-1991-1079009-6.pdf

Http://www.ams.org/journals/mcom/1991-57-195/s0025-5718-1991-1079009-6/s0025-5718-1991-1079009-6.pdf

|journal=Mathematics of Computation

| journal = 计算数学

|date=July 1991

日期 = 1991年7月

|pages=435–439

| 页数 = 435-439

|volume=57

57


|title=A Precise Calculation of the Feigenbaum Constants

费根鲍姆常数的精确计算

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. 6.0 6.1 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

第一名: 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

|last1=Broadhurst

1 = Broadhurst

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

|url=http://www.plouffe.fr/simon/constants/feigenbaum.txt

Http://www.plouffe.fr/simon/constants/feigenbaum.txt

|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

日期 = 1999年3月22日. line feed character in |date= at position 14 (help); |first= missing |last= (help); Check date values in: |date= (help); Missing or empty |title= (help)

}}

|last=Briggs

|url=http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf

|publisher=University of Melbourne

|year=1997

|degree=PhD

|title=Feigenbaum scaling in discrete dynamical systems

}}


Category:Dynamical systems

类别: 动力系统

Category:Mathematical constants

类别: 数学常数


Category:Bifurcation theory

类别: 分岔理论

External links

Category:Chaos theory

范畴: 混沌理论


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