Feigenbaum Map Research Articles

Elementary explanation of boundary shading in chaotic-attractor plots for the Feigenbaum map and the circle map.

An elementary explanation is given for the boundary shadings observed in plots of chaotic attractors for the Feigenbaum map and the circle map. An explicit analytic representation of these phenomena is presented which provides the location of the boundaries and which side of the boundaries is shaded. For the Feigenbaum map, the same analytic expressions also yield the U sequence of periodic windows and their locations in the chaotic zone.

Generalized dimensions of strange attractors

It is pointed out that there exists an infinity of generalized dimensions for strange attractors, related to the order- q Renyi entropies. They are monotonically decreasing with q. For q = 0, 1 and 2, they are the capacity, the information dimension, and the correlation exponent, respectively. For all q, they are measurable from recurrence times in a time series, without need for a box-counting algorithm. For the Feigenbaum map and for the generalized Baker transformation, all generalized dimensions are finite and calculable, and depend non-trivially on q.

Attracting Cantor set of positive measure for a C∞ map of an interval

AbstractWe give an example of a smooth map of an interval into itself, conjugate to the Feigenbaum map, for which the attracting Cantor set has positive Lebesgue measure.

A note on the power spectrum of the iterates of Feigenbaum's function

We give rigorous bounds on the scaling behaviour of the power spectrum for Feigenbaum's map.