Feigenbaum function

Abstract: For realλ a correspondence is made between the Julia setBλ forz→(z−λ)2, in the hyperbolic case, and the set ofλ-chainsλ±√(λ±√(λ±..., with the aid of Cremer's theorem. It is shown how a number of features ofBλ can be understood in terms ofλ-chains. The structure ofBλ is determined by certain equivalence classes ofλ-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics ofλ-chains. The functional equations obeyed by attractive cycles are investigated, and their relation toλ-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets andλ-chains. Certain “Julia sets” associated with the Feigenbaum function and some theorems of Lanford are discussed.

Abstract: The universal map for the period-doubling transition to chaos is studied numerically in the complex plane. The boundary of the domain of analyticity of this function is obtained graphically and is shown to be a fractal with self-similar properties obtained by rescaling with the universal constantsα andδ. In the complex parameter plane, this domain is shown asymptotically to be similar to part of the Mandelbrot set.

Abstract: The paper shows the results of computer simulations which were performed with the use of OCaml functional language. The simulations show Feigenbaum trees for a broad spectrum of functions. The possibility to magnify selected areas of the generated fractals permits viewing a whole diversity of structures which are invisible on a normal scale. What is most important, however, is that for non-Feigenbaum functions, the Author discovered the structures which differ completely in terms of quality from those of classic Feigenbaum trees.