Feigenbaum's First Constant

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

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

It is given by the limit[1]

where an are discrete values of a at the nth period doubling.

Names

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  • Feigenbaum constant
  • Feigenbaum bifurcation velocity
  • delta

Value

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  • 30 decimal places : δ = 4.669201609102990671853203820466
  • (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 0.0047%

Illustration

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Non-linear maps

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

 

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:[2]

n Period Bifurcation parameter (an) Ratio an−1an−2/anan−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

 

with real parameter a and variable x. Tabulating the bifurcation values again:[3]

n Period Bifurcation parameter (an) Ratio an−1an−2/anan−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 4.6680
8 256 3.5699340 4.6768

Fractals

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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

 

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

n Period = 2n Bifurcation parameter (cn) Ratio  
1 2 −0.75
2 4 −1.25
3 8 −1.3680989 4.2337
4 16 −1.3940462 4.5515
5 32 −1.3996312 4.6459
6 64 −1.4008287 4.6639
7 128 −1.4010853 4.6668
8 256 −1.4011402 4.6740
9 512 −1.401151982029 4.6596
10 1024 −1.401154502237 4.6750
... ... ... ...
−1.4011551890...

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.

 
Julia set for the Feigenbaum point

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.

References

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  1. ^ Jordan, D. W.; Smith, P. (2007). Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th ed.). Oxford University Press. ISBN 978-0-19-920825-8.
  2. ^ Alligood, p. 503.
  3. ^ Alligood, p. 504.