Jordan & Smith Mathematical Techniques 4e Pdf

Mathematical constants related to chaotic behavior

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 physicist Mitchell J. Feigenbaum.

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. Feigenbaum made this discovery in 1975,^{[1] [2]} and he officially published it in 1978.^[3]

The first constant [edit]

The first Feigenbaum constant δ 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^[4]

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

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

Names [edit]

• Feigenbaum bifurcation velocity
• delta

Value [edit]

• 30 decimal places : δ = 4.669201 609 102 990 671 853 203 820 466 …
• (sequence A006890 in the OEIS)
• A simple rational approximation is: 621 / 133 , which is correct to 5 significant values (when rounding). For more precision use 1228 / 263 , which is correct to 7 significant values.
• Is approximately equal to 10( 1 / π − 1 ), with an error of 0.0015%

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 a ₁ , a ₂ etc. These are tabulated below:^[5]

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

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

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

with real parameter a and variable x . Tabulating the bifurcation values again:^[6]

n	Period	Bifurcation parameter ( an )	Ratio a n−1 − a n−2 / a n − a n−1
1	2	3	—
2	4	3.4494897	—
3	8	3.5440903	4.7514
4	16	3.5644073	4.6562
5	32	3.5687594	4.6683
6	64	3.5696916	4.6686
7	128	3.5698913	4.6692
8	256	3.5699340	4.6694

Fractals [edit]

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.

In the case of the Mandelbrot set for complex quadratic polynomial

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

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).

n	Period = 2 n	Bifurcation parameter ( cn )	Ratio ${\displaystyle ={\dfrac {c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}}}$
1	2	−0.75	—
2	4	−1.25	—
3	8	−1.3680989	4.2337
4	16	−1.3940462	4.5515
5	32	−1.3996312	4.6458
6	64	−1.4008287	4.6639
7	128	−1.4010853	4.6682
8	256	−1.4011402	4.6689
9	512	−1.401151 982 029
10	1024	−1.401154 502 237
∞		−1.401155 1890 …

Bifurcation parameter is a root point of period-2 ⁿ component. This series converges to the Feigenbaum point c = −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.

The second constant [edit]

The second Feigenbaum constant or Feigenbaum's alpha constant (sequence A006891 in the OEIS),

${\displaystyle \alpha =2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218...,}$

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.^[7]

These numbers apply to a large class of dynamical systems (for example, dripping faucets to population growth).^[7]

A simple rational approximation is 13 / 11 × 17 / 11 × 37 / 27 = 8177 / 3267 .

Properties [edit]

Both numbers are believed to be transcendental, although they have not been proven to be so.^[8] 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—with computer-assistance—in 1982^[9] (with a small correction by Jean-Pierre Eckmann and Peter Wittwer of the University of Geneva in 1987^[10]). 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.^[11]

See also [edit]

• Bifurcation diagram
• Bifurcation theory
• Cascading failure
• Feigenbaum function
• List of chaotic maps

Notes [edit]

1. ^ Feigenbaum, M. J. (1976). "Universality in complex discrete dynamics" (PDF). Los Alamos Theoretical Division Annual Report 1975–1976.
2. ^ Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996). Chaos: An Introduction to Dynamical Systems. Springer. ISBN0-387-94677-2.
3. ^ Feigenbaum, Mitchell J. (1978). "Quantitative universality for a class of nonlinear transformations". Journal of Statistical Physics. 19 (1): 25–52. doi:10.1007/BF01020332. S2CID 124498882.
4. ^ Jordan, D. W.; Smith, P. (2007). Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th ed.). Oxford University Press. ISBN978-0-19-920825-8.
5. ^ Alligood, p. 503.
6. ^ Alligood, p. 504.
7. ^ ^{a b} Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos. Studies in Nonlinearity. Perseus Books. ISBN978-0-7382-0453-6.
8. ^ Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
9. ^ 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.
10. ^ 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. S2CID 121353606.
11. ^ 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. S2CID 119594350.

References [edit]

• Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences Springer, 1996, ISBN 978-0-38794-677-1
• Briggs, Keith (July 1991). "A Precise Calculation of the Feigenbaum Constants" (PDF). Mathematics of Computation. 57 (195): 435–439. Bibcode:1991MaCom..57..435B. doi:10.1090/S0025-5718-1991-1079009-6.
• Briggs, Keith (1997). Feigenbaum scaling in discrete dynamical systems (PDF) (PhD thesis). University of Melbourne.
• Broadhurst, David (22 March 1999). "Feigenbaum constants to 1018 decimal places".

• Weisstein, Eric W. "Feigenbaum Constant". MathWorld.

External links [edit]

• Feigenbaum Constant – from Wolfram MathWorld
• OEIS sequence A006890 (Decimal expansion of Feigenbaum bifurcation velocity)

OEIS sequence A006891 (Decimal expansion of Feigenbaum reduction parameter)

OEIS sequence A094078 (Decimal expansion of Pi + arctan(e^Pi))

• Feigenbaum constant – PlanetMath
• Moriarty, Philip; Bowley, Roger (2009). " δ – Feigenbaum Constant". Sixty Symbols. Brady Haran for the University of Nottingham.

Jordan & Smith Mathematical Techniques 4e Pdf

Source: https://en.wikipedia.org/wiki/Feigenbaum_constants

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