In the study of
dynamical systems the term
Feigenbaum function has been used to describe two different functions introduced by the physicist
Mitchell Feigenbaum:
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′(0) = 0, and
- g′′(0) < 0
For a particular form of solution with a quadratic dependence of the solutionnear x=0, the inverse
1/λ=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<sub>n</sub>. For a fixed
d<sub>n</sub> the set of segments forms a cover
Δ<sub>n</sub> of the attractor. The ratio of segments from two...
Read More