Feigenbaum Constant

A universal constant for functions approaching Chaos via period doubling. It was discovered by Feigenbaum in 1975 and demonstrated rigorously by Lanford (1982) and Collet and Eckmann (1979, 1980). The Feigenbaum constant $\delta$ characterizes the geometric approach of the bifurcation parameter to its limiting value. Let $\mu_k$ be the point at which a period $2^k$ cycle becomes unstable. Denote the converged value by $\mu_\infty$. Assuming geometric convergence, the difference between this value and $\mu_k$ is denoted

$$\lim_{k\to\infty} \mu_\infty -\mu_k = {\Gamma\over\delta^k},$$ (1)

where $\Gamma$ is a constant and $\delta$ is a constant $>1$. Solving for $\delta$ gives

$$\delta \equiv \lim_{n\to\infty} {\mu_{n+1}-\mu_n\over \mu_{n+2}-\mu_{n+1}}$$ (2)

(Rasband 1990, p. 23). For the Logistic Equation,

$$\delta = 4.6692016091\ldots$$ (3)

$$\Gamma = 2.637\ldots$$ (4)

$$\mu_\infty = 3.5699456\ldots.$$ (5)

Amazingly, the Feigenbaum constant $\delta\approx 4.669$ is ``universal'' (i.e., the same) for all 1-D Maps $f(x)$ if $f(x)$ has a single locally quadratic Maximum. More specifically, the Feigenbaum constant is universal for 1-D Maps if the Schwarzian Derivative

$$D_{\rm Schwarzian} \equiv {f'''(x)\over f'(x)} -{3\over 2}\left[{f''(x)\over f'(x)}\right]^2$$ (6)

is Negative in the bounded interval (Tabor 1989, p. 220). Examples of maps which are universal include the Hénon Map, Logistic Map, Lorenz System, Navier-Stokes truncations, and sine map $x_{n+1}=a\sin(\pi x_n)$. The value of the Feigenbaum constant can be computed explicitly using functional group renormalization theory. The universal constant also occurs in phase transitions in physics and, curiously, is very nearly equal to

$$\pi+\tan^{-1} (e^\pi) = 4.669201932\ldots.$$ (7)

The Circle Map is not universal, and has a Feigenbaum constant of $\delta\approx 2.833$. For an Area-Preserving 2-D Map with

$$x_{n+1} = f(x_n,y_n)$$ (8)

$$y_{n+1} = g(x_n,y_n),$$ (9)

the Feigenbaum constant is $\delta=0.7210978\ldots$ (Tabor 1989, p. 225). For a function of the form

$$f(x)=1-a\vert x\vert^n$$ (10)

with $a$ and $n$ constant and $n$ an Integer, the Feigenbaum constant for various $n$ is given in the following table (Briggs 1991, Briggs et al. 1991), which updates the values in Tabor (1989, p. 225).

$n$	$\delta$
2	5.9679
4	7.2846
6	8.3494
8	9.2962

An additional constant $\alpha$, defined as the separation of adjacent elements of Period Doubled Attractors from one double to the next, has a value

$$\lim_{n\to\infty} {d_n\over d_{n+1}} \equiv -\alpha =-2.502907875\ldots$$ (11)

for ``universal'' maps (Rasband 1990, p. 37). This value may be approximated from functional group renormalization theory to the zeroth order by

$$1-\alpha^{-1}={1-\alpha^{-2}\over [1-\alpha ^{-2}(1-\alpha ^{-1})]^2},$$ (12)

which, when the Quintic Equation is numerically solved, gives $\alpha= -2.48634\dots$, only 0.7% off from the actual value (Feigenbaum 1988).

See also Attractor, Bifurcation, Feigenbaum Function, Linear Stability, Logistic Map, Period Doubling

References

Briggs, K. ``A Precise Calculation of the Feigenbaum Constants.'' Math. Comput. 57, 435-439, 1991.

Briggs, K.; Quispel, G.; and Thompson, C. ``Feigenvalues for Mandelsets.'' J. Phys. A: Math. Gen. 24 3363-3368, 1991.

Briggs, K.; Quispel, G.; and Thompson, C. ``Feigenvalues for Mandelsets.'' http://epidem13.plantsci.cam.ac.uk/~kbriggs/.

Collet, P. and Eckmann, J.-P. ``Properties of Continuous Maps of the Interval to Itself.'' Mathematical Problems in Theoretical Physics (Ed. K. Osterwalder). New York: Springer-Verlag, 1979.

Collet, P. and Eckmann, J.-P. Iterated Maps on the Interval as Dynamical Systems. Boston, MA: Birkhäuser, 1980.

Eckmann, J.-P. and Wittwer, P. Computer Methods and Borel Summability Applied to Feigenbaum's Equations. New York: Springer-Verlag, 1985.

Feigenbaum, M. J. ``Presentation Functions, Fixed Points, and a Theory of Scaling Function Dynamics.'' J. Stat. Phys. 52, 527-569, 1988.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html

Finch, S. ``Generalized Feigenbaum Constants.'' http://www.mathsoft.com/asolve/constant/fgnbaum/general.html.

Lanford, O. E. ``A Computer-Assisted Proof of the Feigenbaum Conjectures.'' Bull. Amer. Math. Soc. 6, 427-434, 1982.

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, 1990.

Stephenson, J. W. and Wang, Y. ``Numerical Solution of Feigenbaum's Equation.'' Appl. Math. Notes 15, 68-78, 1990.

Stephenson, J. W. and Wang, Y. ``Relationships Between the Solutions of Feigenbaum's Equations.'' Appl. Math. Let. 4, 37-39, 1991.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.