HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
bifurcation theory Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Mo ...
, the Feigenbaum constants are two
mathematical constants A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
which both express ratios in a
bifurcation diagram In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the syst ...
for a non-linear map. They are named after the physicist
Mitchell J. Feigenbaum Mitchell Jay Feigenbaum (December 19, 1944 – June 30, 2019) was an American mathematical physics, mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants. Early life Feigenbaum was bor ...
.


History

Feigenbaum originally related the first constant to the
period-doubling bifurcation In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. W ...
s in 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 ...
, but also showed it to hold for all
one-dimensional In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the ...
maps A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
with a single quadratic
maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
. As a consequence of this generality, every
chaotic system Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
that corresponds to this description will bifurcate at the same rate. Feigenbaum made this discovery in 1975, and he officially published it in 1978.


The first constant

The first Feigenbaum constant is the limiting
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of each bifurcation interval to the next between every period doubling, of a one-
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
map :x_ = f(x_i), where is a function parameterized by the bifurcation parameter . It is given by the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
:\delta = \lim_ \frac = 4.669\,201\,609\,\ldots, where are discrete values of at the th period doubling.


Names

* Feigenbaum Constant * Feigenbaum bifurcation velocity * delta


Value

* 30 decimal places : = * * A simple rational approximation is: , which is correct to 5 significant values (when rounding). For more precision use , which is correct to 7 significant values. * Is approximately equal to , with an error of 0.0015%


Illustration


Non-linear maps

To see how this number arises, consider the real one-parameter map :f(x)=a-x^2. Here is the bifurcation parameter, is the variable. The values of for which the period doubles (e.g. the largest value for with no period-2 orbit, or the largest with no period-4 orbit), are , etc. These are tabulated below: : The ratio in the last column converges to the first Feigenbaum constant. The same number arises for 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 ...
: f(x) = a x (1 - x) with real parameter and variable . Tabulating the bifurcation values again: :


Fractals

In the case of the
Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This ...
for
complex quadratic polynomial A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers. Properties Quadratic polynomials have the following properties, regardless of the form: *It is a unicritical polynomial, i.e. it has one ...
: f(z) = z^2 + c the Feigenbaum constant is the ratio between the diameters of successive circles on the
real axis In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poi ...
in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
(see animation on the right). : Bifurcation parameter is a root point of period- component. This series converges to the Feigenbaum point = −1.401155...... The ratio in the last column converges to the first Feigenbaum constant. Other maps also reproduce this ratio, in this sense the Feigenbaum constant in bifurcation theory is analogous to in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
.


The second constant

The second Feigenbaum constant or Feigenbaum's alpha constant , :\alpha = 2.502\,907\,875\,095\,892\,822\,283\,902\,873\,218..., is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold). A negative sign is applied to when the ratio between the lower subtine and the width of the tine is measured. These numbers apply to a large class of
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s (for example, dripping faucets to population growth). A simple rational approximation is × × = .


Properties

Both numbers are believed to be transcendental, although they have not been proven to be so. There is also no known proof that either constant is irrational. The first proof of the universality of the Feigenbaum constants was carried out by
Oscar Lanford Oscar Eramus Lanford III (January 6, 1940 – November 16, 2013) was an American mathematician working on mathematical physics and dynamical systems theory. Professional career Born in New York, Lanford was awarded his undergraduate degree ...
—with computer-assistance—in 1982 (with a small correction by
Jean-Pierre Eckmann Jean-Pierre Eckmann (born 27 January 1944) is a Swiss mathematical physicist in the department of theoretical physics at the University of Geneva and a pioneer of chaos theory and social network analysis.. Eckmann is the son of mathematician Be ...
and Peter Wittwer of the
University of Geneva The University of Geneva (French: ''Université de Genève'') is a public research university located in Geneva, Switzerland. It was founded in 1559 by John Calvin as a theological seminary. It remained focused on theology until the 17th centu ...
in 1987). Over the years, non-numerical methods were discovered for different parts of the proof, aiding
Mikhail Lyubich Mikhail Lyubich (born 25 February 1959 in Kharkiv, Ukraine) is a mathematician who made important contributions to the fields of holomorphic dynamics and chaos theory. Lyubich graduated from Kharkiv University with a master's degree in 1980, and ...
in producing the first complete non-numerical proof.


See also

*
Bifurcation diagram In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the syst ...
*
Bifurcation theory Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Mo ...
*
Cascading failure A cascading failure is a failure in a system of interconnected parts in which the failure of one or few parts leads to the failure of other parts, growing progressively as a result of positive feedback. This can occur when a single part fails, i ...
* Feigenbaum function *
List of chaotic maps In mathematics, a chaotic map is a map (namely, an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functi ...


Notes


References

* Alligood, Kathleen T., Tim D. Sauer, James A. Yorke, ''Chaos: An Introduction to Dynamical Systems, Textbooks in mathematical sciences'' Springer, 1996, * * * *


External links


Feigenbaum Constant – from Wolfram MathWorld
* : :
Feigenbaum constant
– PlanetMath * * {{DEFAULTSORT:Feigenbaum Constants Dynamical systems Mathematical constants Bifurcation theory Chaos theory