Feigenbaum function

Feigenbaum Function

Feigenbaum function

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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&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<sub>n</sub>. For a fixed d<sub>n</sub> the set of segments forms a cover &Delta;<sub>n</sub> of the attractor. The ratio of segments from two...
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