Feigenbaum Constant

The Feigenbaum constant 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 is increased for fixed . The plot above is made by iterating equation (1) with several hundred times for a series of discrete but closely spaced values of , 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 as a function of . The plot above (Trott, pers. comm.) shows the resulting curves for , 2, and 4.

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

where is a constant and is a constant now known as the Feigenbaum constant. Solving for gives

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

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

For equation (1) with , the onsets of bifurcations occur at , 1.25, 1.368099, 1.39405, 1.399631, ..., giving convergents to for , 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 is known as the Feigenbaum constant and is the associated "reduction parameter."

Briggs (1991) calculated 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 is algebraic, or if it can be expressed in terms of other mathematical constants (Borwein and Bailey 2003, p. 53).

Briggs (1991) calculated 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 and associated reduction parameter are "universal" for all one-dimensional maps if has a single locally quadratic maximum. This was conjecture by Feigenbaum, and demonstrated rigorously by Lanford (1982) for the case , and by Epstein (1985) for all .

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 . 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 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 , only 0.7% off from the actual value (Feigenbaum 1988).

For an area-preserving two-dimensional map with

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

For a function of the form (1), the Feigenbaum constant for various 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 and be even functions with and

and as large as possible. Let be positive numbers with

and as small as possible. Also let be the order of the nearest singularity, with

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