Feigenbaum Constant


The Feigenbaum constant delta is a universal constant for functions approaching chaos via period doubling. It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function


and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter mu is increased for fixed x. The plot above is made by iterating equation (1) with r=2 several hundred times for a series of discrete but closely spaced values of mu, discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining.


A similar plot that more directly shows the cycle may be constructed by plotting f^n(x)-x as a function of mu. The plot above (Trott, pers. comm.) shows the resulting curves for n=1, 2, and 4.

Let mu_n be the point at which a period 2^n-cycle appears, and denote the converged value by mu_infty. Assuming geometric convergence, the difference between this value and mu_n is denoted


where Gamma is a constant and delta>1 is a constant now known as the Feigenbaum constant. Solving for delta gives


(Rasband 1990, p. 23; Briggs 1991). An additional constant alpha, defined as the separation of adjacent elements of period doubled attractors from one double to the next, has a value


where d_n is the value of the nearest cycle element to 0 in the 2^n cycle (Rasband 1990, p. 37; Briggs 1991).

For equation (1) with r=2, the onsets of bifurcations occur at mu=0.75, 1.25, 1.368099, 1.39405, 1.399631, ..., giving convergents to delta for n=1, 2, 3, ... of 4.23374, 4.5515, 4.64617, ....

For the logistic map,


(OEIS A006890, A098587, and A006891; Broadhurst 1999; Wolfram 2002, p. 920), where delta is known as the Feigenbaum constant and alpha is the associated "reduction parameter."

Briggs (1991) calculated delta to 84 digits, Briggs (1997) to 576 decimal places (of which 344 were correct), and Broadhurst (1999) to 1018 decimal places. It is not known if the Feigenbaum constant delta is algebraic, or if it can be expressed in terms of other mathematical constants (Borwein and Bailey 2003, p. 53).

Briggs (1991) calculated alpha to 107 digits, Briggs (1997) to 576 decimal places (of which 346 were correct), and Broadhurst (1999) to 1018 decimal places.

Amazingly, the Feigenbaum constant delta and associated reduction parameter alpha are "universal" for all one-dimensional maps f(x) if f(x) has a single locally quadratic maximum. This was conjecture by Feigenbaum, and demonstrated rigorously by Lanford (1982) for the case r=2, and by Epstein (1985) for all r<14.

More specifically, the Feigenbaum constant is universal for one-dimensional maps if the Schwarzian derivative


is negative in the bounded interval (Tabor 1989, p. 220). Examples of maps which are universal include the Hénon map, logistic map, Lorenz attractor, Navier-Stokes truncations, and sine map x_(n+1)=asin(pix_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.

The value of alpha for a universal map may be approximated from functional group renormalization theory to the zeroth order by solving


which can be rewritten as the quintic equation


Solving numerically for the smallest real root gives alpha=-2.48634..., only 0.7% off from the actual value (Feigenbaum 1988).

For an area-preserving two-dimensional map with


the Feigenbaum constant is delta=8.7210978... (Tabor 1989, p. 225).

For a function of the form (1), the Feigenbaum constant for various r is given in the following table (Briggs 1991, Briggs et al. 1991, Finch 2003), which updates the values in Tabor (1989, p. 225).


Broadhurst (1999) considered additional Feigenbaum constants. Let g(x) and f(x) be even functions with g(0)=f(0)=1 and


and delta as large as possible. Let (b,c,d) be positive numbers with


and (b,c^2+d^2) as small as possible. Also let kappa be the order of the nearest singularity, with


as z tends to zero. The values of these constants are summarized in the following table.


See also

Attractor, Bifurcation, Feigenbaum Constant Approximations, Feigenbaum Function, Linear Stability, Logistic Map, Period Doubling, Tent Map

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Feigenbaum Constant

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Weisstein, Eric W. "Feigenbaum Constant." From MathWorld--A Wolfram Web Resource.

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