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

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

It is given by the limit
 * $$\delta = \lim_{n \to \infty} \frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}} = 4.669\,201\,609\,\ldots,$$

where $a_{n}$ are discrete values of $a$ at the $n$th period doubling.

Names

 * Feigenbaum constant
 * Feigenbaum bifurcation velocity
 * delta

Value

 * 30 decimal places : $δ$ = 4.669 201  609  102  990  671  853  203  820  466  …
 * 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%
 * Is approximately equal to $a$, with an error of 0.0047%

Non-linear maps
To see how this number arises, consider the real one-parameter map
 * $$f(x)=a-x^2.$$

Here $x$ is the bifurcation parameter, $a$ 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 $a_{1}$ with no period-4 orbit), are $a_{2}$, $n$ etc. These are tabulated below:


 * {| class="wikitable"

! $a_{n}$ ! Period ! Bifurcation parameter ($an−1 − an−2⁄an − an−1$) ! Ratio $a$
 * 1
 * 2
 * 0.75
 * 2
 * 4
 * 1.25
 * 3
 * 8
 * 4.2337
 * 4
 * 16
 * 4.5515
 * 5
 * 32
 * 4.6458
 * 6
 * 64
 * 4.6639
 * 7
 * 128
 * 4.6682
 * 8
 * 256
 * 4.6689
 * }
 * 6
 * 64
 * 4.6639
 * 7
 * 128
 * 4.6682
 * 8
 * 256
 * 4.6689
 * }
 * 8
 * 256
 * 4.6689
 * }
 * 4.6689
 * }
 * }

The ratio in the last column converges to the first Feigenbaum constant. The same number arises for the logistic map
 * $$ f(x) = a x (1 - x) $$

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


 * {| class="wikitable"

! $a_{n}$ ! Period ! Bifurcation parameter ($an−1 − an−2⁄an − an−1$) ! Ratio $x$
 * 1
 * 2
 * 3
 * 2
 * 4
 * 3
 * 8
 * 4.7514
 * 4
 * 16
 * 4.6562
 * 5
 * 32
 * 4.6683
 * 6
 * 64
 * 4.6686
 * 7
 * 128
 * 4.6680
 * 8
 * 256
 * 4.6768
 * }
 * 4.6683
 * 6
 * 64
 * 4.6686
 * 7
 * 128
 * 4.6680
 * 8
 * 256
 * 4.6768
 * }
 * 8
 * 256
 * 4.6768
 * }
 * 4.6768
 * }
 * }

Fractals


In the case of the Mandelbrot set for complex quadratic polynomial
 * $$ f(z) = z^2 + c $$

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).


 * {| class="wikitable"

! $n$ ! Period = $2^{n}$ ! Bifurcation parameter ($c_{n}$) ! Ratio $$= \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} $$
 * 1
 * 2
 * 2
 * 4
 * 3
 * 8
 * 4.2337
 * 4
 * 16
 * 4.5515
 * 5
 * 32
 * 4.6459
 * 6
 * 64
 * 4.6639
 * 7
 * 128
 * 4.6668
 * 8
 * 256
 * 4.6740
 * 9
 * 512
 * 4.6596
 * 10
 * 1024
 * 4.6750
 * }
 * 4.6639
 * 7
 * 128
 * 4.6668
 * 8
 * 256
 * 4.6740
 * 9
 * 512
 * 4.6596
 * 10
 * 1024
 * 4.6750
 * }
 * 4.6596
 * 10
 * 1024
 * 4.6750
 * }
 * 4.6750
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }
 * }

Bifurcation parameter is a root point of period-$∞$ component. This series converges to the Feigenbaum point $2^{n}$ = −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 $\pi$ in geometry and $c$ in calculus.