In the study of
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
the term Feigenbaum function has been used to describe two different functions introduced by the physicist
Mitchell Feigenbaum
Mitchell Jay Feigenbaum (December 19, 1944 – June 30, 2019) was an American mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants.
Early life
Feigenbaum was born in Philadelphia, Pe ...
:
* the solution to the Feigenbaum-Cvitanović functional equation; and
* the scaling function that described the covers of the
attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
of the
logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
Feigenbaum-Cvitanović functional equation
This functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. Discovered by
Mitchell Feigenbaum
Mitchell Jay Feigenbaum (December 19, 1944 – June 30, 2019) was an American mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants.
Early life
Feigenbaum was born in Philadelphia, Pe ...
and
Predrag Cvitanović
Predrag Cvitanović (; born April 1, 1946) is a theoretical physicist regarded for his work in nonlinear dynamics, particularly his contributions to periodic orbit theory.
Life
Cvitanović earned his B.S. from MIT in 1969 and his Ph.D. at Cornel ...
,
[Footnote on p. 46 of Feigenbaum (1978) states "This exact equation was discovered by P. Cvitanović during discussion and in collaboration with the author."] the equation is the mathematical expression of the
universality of period doubling. It specifies a function ''g'' and a parameter by the relation
:
with the initial conditions
* ''g''(0) = 1,
* ''g''′(0) = 0, and
* ''g''′′(0) < 0
For a particular form of solution with a quadratic dependence of the solution
near is one of the
Feigenbaum constant
In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum.
Hist ...
s.
Scaling function
The Feigenbaum scaling function provides a complete description of the
attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
of the
logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
at the end of the period-doubling cascade. The attractor is a
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Thr ...
, 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
n''. For a fixed ''d
n'' the set of segments forms a cover ''Δ
n'' of the attractor. The ratio of segments from two consecutive covers, ''Δ
n'' and ''Δ
n+1'' can be arranged to approximate a function ''σ'', the Feigenbaum scaling function.
See also
*
Logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
*
Presentation function
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
Notes
Bibliography
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* Bound as ''Order in Chaos, Proceedings of the International Conference on Order and Chaos held at the Center for Nonlinear Studies, Los Alamos, New Mexico 87545,USA 24–28 May 1982'', Eds. David Campbell, Harvey Rose; North-Holland Amsterdam .
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* {{MathWorld, urlname=FeigenbaumFunction, title=Feigenbaum Function
Chaos theory
Dynamical systems