Feigenbaum constants

The two Feigenbaum constants α and δ are mathematical constants, which play an important role in the chaos research.

Research

The numerical value of δ was first published by the physicists Siegfried Grossmann and Stefan Thomae 1977. Mitchell Feigenbaum, who discovered in 1975 that figure in the study of fixed points of iterated functions, published in 1978 a paper on the universality of this constant. The significance of these constants for the chaos theory is often compared with that of π for the geometry. Are your numbers

These figures appear to show depending on a parameter regular or chaotic behavior in the context of non-linear systems. The transition to chaos is characterized by a parameter range with an oscillating behavior. For chaotic area towards this takes the oscillation gradually by a factor of two to a phenomenon that is known as period doubling. The associated parameter intervals with increasing period shorter. The ratio of the lengths of successive intervals of different parameters period thereby seeks δ against the Feigenbaum constant.

For the case of nonlinear systems, which are represented by sequences of numbers with non-linear recursive Education Act and the show depending on a parameter such behavior, this phenomenon is the so-called Feigenbaum diagram can be used. It turns followers depending on this parameter is in fact from a sequence index, after which the sequence has stabilized to a certain behavior, such as convergence to a limit cycle or chaotic behavior, and thus corresponds to a representation of the accumulation points of the sequence. Points at which there is a period doubling, are characterized by a fork -like structures, referred to as bifurcations. The ratio of the widths of consecutive forks closest bifurcation point α thereby strives against the Feigenbaum constant. It is often referred to as second Feigenbaum constant.

In the region of chaotic behavior occur on islands periodic behavior. The transition of the behavior in these islands to the main area not chaotic behavior toward is instantaneous, in the other direction it is again characterized by period doublings, quantitatively show the same behavior.

This qualitative behavior and the related ratios do not depend on the details of the mathematical or physical nonlinear system from, but represent a universal and thus fundamental law of such systems Represents the simplest mathematical example is the behavior of sequences of numbers square recursive Education Act as the logistic equation and the sequence of numbers that is the Mandelbrot set is based.

It is assumed that δ and α are transcendent, a corresponding proof is still pending. Keith Briggs developed and used in 1991 a method for calculating the constant with high numerical accuracy. The most accurate values ​​with 1018 decimal places were given by David Broadhurst 1999.