Fig tree constant

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

exploration

The numerical value of was first published in 1977 by the physicists Siegfried Großmann and Stefan Thomae . Mitchell Feigenbaum , who had discovered this number in 1975 while studying the fixed points of iterated functions, published a paper in 1978 on the universality of this constant. The importance of these constants for chaos research is often compared to that of π for geometry. Their numerical values ​​are

(Follow A006890 in OEIS ),
(Follow A006891 in OEIS ).
Excerpt from the fig tree diagram of the logistic equation from the area of ​​the bifurcations at the transition from order (left) to chaos (right).

These numbers appear in connection with non-linear systems , which show regular or chaotic behavior depending on a parameter . The transition into chaos is characterized by a parameter area with oscillating behavior. Towards the chaotic area, the oscillation period increases step by step by a factor of two, a phenomenon known as period doubling. The associated parameter intervals become shorter and shorter as the period increases. The ratio of the lengths of successive parameter intervals of different periods tends towards the Feigenbaum constant δ.

In the case of non-linear systems, which are represented by number sequences with a non-linear recursive formation law and which show such behavior as a function of a parameter, this phenomenon can be shown in the so-called fig tree diagram . It represents sequence elements as a function of this parameter, starting with a sequence index, according to which the sequence has settled on a certain behavior, such as convergence towards a limit cycle or chaotic behavior, and thus corresponds to a representation of the accumulation points of the sequence. Places where the period doubles are characterized by fork-shaped structures known as bifurcations . The ratio of the widths of successive forks at the next bifurcation point tends towards the Feigenbaum constant α. It is often referred to as the second fig tree constant.

In the area of ​​chaotic behavior, islands of periodic behavior appear, "periodic windows". The transition from chaotic behavior to these islands of non-chaotic behavior (from left to right in the diagram) is instantaneous, from the periodic windows the transition is again characterized by period doublings, which quantitatively show the same behavior (self-similarity). Bénard cells are a physical process that can be explained by the fig tree duplication.

This qualitative behavior and the associated numerical relationships do not depend on the details of the mathematical or physical nonlinear system, but rather represent a universal and thus fundamental law of such systems. The simplest mathematical example is the behavior of sequences of numbers of quadratic recursive formation law such as the logistic equation and Sequence of numbers on which the Mandelbrot set is based.

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

literature

Web links

Individual evidence

  1. ↑ Nonlinear Dynamics and Chaos (PDF)
  2. ↑ Keith Briggs . - Keith Briggs's homepage
  3. ↑ Feigenbaum constants to 1018 decimal places . In: Plouffe's Inverter , email of March 22, 1999