Feigenbaum function

In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:

• the solution to the Feigenbaum-Cvitanović functional equation; and
• the scaling function that described the covers of the attractor of the logistic map

Functional equation

The functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. The functional equation is the mathematical expression of the universality of period doubling. The equation is used to specify a function g and a parameter λ by the relation

<math> g(x) = frac g( g(lambda x ) ) </math>

with the boundary conditions

• g(0) = 1,
• g&prime;(0) = 0, and
• g&prime;&prime;(0) < 0

For a particular form of solution with a quadratic dependence of the solutionnear x=0, the inverse 1/&lambda;=2.5029... is one of the Feigenbaum constants.

Scaling function

The Feigenbaum scaling function provides a complete description of the attractor of the logistic map at the end of the period-doubling cascade. The attractor is a Cantor set set, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size d_n. For a fixed d_n the set of segments forms a cover &Delta;_n of the attractor. The ratio of segments from two...
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