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Topic: Feigenbaum constant

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	PlanetMath: Feigenbaum constant
		That is, the ratio of the intervals between the bifurcation points approaches Feigenbaum's constant.
		Feigenabum's constant appears in problems of fluid-flow turbulence, electronic oscillators, chemical reactions, and even the Mandelbrot set (the ``budding'' of the Mandelbrot set along the negative real axis occurs at intervals determined by Feigenbaum's constant).
		This is version 3 of Feigenbaum constant, born on 2002-04-07, modified 2005-02-28.
	planetmath.org /encyclopedia/FeigenbaumConstant.html (187 words)

	Feigenbaum constant - Wikipedia, the free encyclopedia
		The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum.
		Feigenbaum originally related this number to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps displaying a single hump.
		Feigenbaum's constant can be used to predict when chaos will arise in such systems before it ever occurs.
	en.wikipedia.org /wiki/Feigenbaum_constant (175 words)

	Fractal Questions and Answers - Feigenbaum's constant (Site not responding. Last check: 2007-10-21)
		The interpretation of the delta constant is as you approach chaos, each periodic region is smaller than the previous by a factor approaching 4.669...
		Feigenbaum's constant is important because it is the same for any function or system that follows the period-doubling route to chaos and has a one-hump quadratic maximum.
		Feigenbaum says, "Asymptotically, the separation of adjacent elements of period-doubled attractors is reduced by a constant value [alpha] from one doubling to the next".
	www.softlab.ntua.gr /Miscellaneous/faq/fractal/faq-doc-10.html (278 words)

	Mathematical constant - Wikipedia, the free encyclopedia
		A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change.
		Therefore, f(1)/f(0) is a mathematical constant, the constant e.
		Mathematical constants are typically elements of the field of real numbers or complex numbers.
	en.wikipedia.org /wiki/Mathematical_constant (336 words)

	[No title]
		Archimedes Constant, pi +------------------------------------------------------------ The Archimedes Constant, pi pi=3.14159 is the length of a half circle with radius 1.
		Bruns constant +------------------------------------------------------------ Bruns constant is the sum of the reciprocals of all twin primes.
		Pythagoras constant +------------------------------------------------------------ The Pythagoras constant is the square root of 2 x=sqrt2=1.41421....
	www.math.harvard.edu /~knill/sofia/data/constants.txt (647 words)

	PlanetMath: Feigenbaum fractal
		The ``canonical'' Feigenbaum fractal is produced by the logistic map (a simple population model),
		One of the most amazing things about this class of fractals is that the bifurcation intervals are always described by Feigenbaum's constant.
		This is version 3 of Feigenbaum fractal, born on 2002-04-07, modified 2002-04-07.
	planetmath.org /encyclopedia/FeigenbaumFractal.html (170 words)

	Feigenbaum's constant (Site not responding. Last check: 2007-10-21)
		Feigenbaum's constant is important because it is the same for any function or system that follows the period-doubling route to chaos and has a one- hump quadratic maximum.
		Briggs, A precise calculation of the Feigenbaum constants, _Mathematics of Computation_ 57 (1991), pp.
		Feigenbaum, The Universal Metric Properties of Nonlinear Transformations, _J.
	www.faqs.org /faqs/fractal-faq/section-10.html (298 words)

	Numerical Constants - Mathematics & Physics - Numericana
		Planck's constant: The ratio of a photon's energy to its frequency.
		The attribution of this irrational constant to Ramanujan was made by Simon Plouffe, as a monument to a famous 1975 April fools column by Martin Gardner in Scientific American (where it was claimed that the above had been proven to be exactly an integer, as conjectured by Ramanujan in 1914 [sic!]).
		Some other set of independent constants could have been used to define the 7 basic units (for example, a conventional value of the electron's charge could replace the conventional permeability of the vacuum) but the following one was chosen after careful considerations.
	home.att.net /~numericana/answer/constants.htm (4599 words)

	Bifurcation; the moons of Jupiter
		Feigenbaum noticed that the first bifurcation or doubling of an insect population arises when the increase of the fertility factor = (1 + square root 6).
		The expression shows that the Feigenbaum constant relates to the split of the planetary space in two areas: sum of the first 5 planets and sum of the last 4 planets.
		The constant "e", an irrational number approximated to 2.7182818, is currently used in the sciences for the study of transition phenomena (dynamic events).
	www.jufo.freeserve.co.uk /kt4.html (2951 words)

	Diode (Site not responding. Last check: 2007-10-21)
		Feigenbaum's constant is estimated based on the parameters at which the first few bifurcations sets in.
		This number is named after M. Feigenbaum who, in 1978, noted that the ratios of parameter distance between two successive period-doublings approach a universal constant as the periods increase to infinity (Ref 3).
		The Feigenbaum estimates were calculated as follows: (88.26 - 73.81)/(92.13 - 88.26) = 3.7 (Table 1, 4th column), etc. The calculated values, which correspond to the first few terms of the sequence of Eq.
	webusers.physics.umn.edu /~rlua/chaos (916 words)

	The Feigenbaum Scenario as a Model of the Limits of Conscious Information Processing (Site not responding. Last check: 2007-10-21)
		This paper first outlines the Feigenbaum scenario of the period doubling sequence and the dynamics of feedback and iteration in the context of the current debate regarding linear and nonlinear dynamics in psychobiology.
		As mentioned previously the Feigenbaum constant is described as "universal" (Feigenbaum,1980) because the same abrupt change between order and apparent chaos after the fourth bifurcation can be found in many different equations provided that they have a parabolic or "hump" structure.
		This evolving view of the Feigenbaum point in psychology suggests that regime of deterministic chaos in the brain-mind may serve as a model of what is called the unconscious.
	www.ernestrossi.com /feigenbaumModelConscious.htm (3947 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Briggs, A precise calculation of the Feigenbaum constants, Mathematics of Computation 57 (1991), pp.
		Feigenbaum, The Universal Metric Properties of Nonlinear Transformations, J. Stat.
		Feigenbaum, Universal Behaviour in Nonlinear Systems, Los Alamos Sci 1 (1980), pp.
	www.math.niu.edu /~rusin/known-math/99/feigen (402 words)

	[No title]
		Feigenbaum constants to 1018 decimal places From: David Broadhurst 22-Mar-1999 To: Simon Plouffe CC: Keith Briggs, David Bailey, Steven Finch Subj: Feigenbaum constants to 1018 decimal places Keith Briggs' thesis gives values of alpha and delta to 576 decimal places.
		I found that his values are good to 346 and 344 places, respectively.
		The Feigenbaum zero nearest to the origin was located to 1018 places.
	pi.lacim.uqam.ca /piDATA/feigenbaum.txt (295 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Concretely, you can see this p as the function that gives a population p(x) at generation n+1 depending on the population x at generation n.
		This constant is always the same (you can modify the equation for p(x), you can change initial population).
		On this subject, I'd have a question : Does anyone know if their is a way to calculate this feigenbaum constant in such a way : K=SUM(i=0 to infinity, Ui), where a general term for Ui is known.
	www.math.niu.edu /~rusin/known-math/96/iterates.chaos (450 words)

	1.3 Universality
		The the period-doubling route to chaos and the constants "alpha" and "delta" appeared in an unruly mess of equations used to describe hydrodynamic flow (Hofstadter).
		The realization that a set of five coupled differential equations describing turbulence could exhibit the same fundamental behavior as the one-dimensional map of the parabola on to itself was one of the great breakthroughs in late twentieth century mathematics.
		The statement was made that the behavior of this system is typical for "any smooth, one-dimensional, non-monotonic function when mapped on to itself." Many books on chaos mention how Feigenbaum's constants and the period-doubling route to chaos appear in other one-dimensional mappings, but few provide examples.
	www.story.com /math/cfd.1.3.html (942 words)

	Numbers, Magick & Motion 3 [continued]
		If the rubidium atom, as Haeglin states, is just the sum of infinite "ideas" in an "intelligent matrix" and these infinite ideas can be thought of as organized actions controlled by the laws of harmonics then a connection to a harmonic series should be evident.
		For the rubidium atom Nature uses the all-encompassing constant of harmonics, Euler's gamma constant.
		The physical property of the rubidium atom can shape the Euler constant into a physical three-dimensional existence as a spherical volume because of a strange mathematical phenomena associated with the factors of the geometric spherical volume formula:
	www.gnostics.com /numbersIIIc.html (1426 words)

	LAB #4 (Site not responding. Last check: 2007-10-21)
		We have also seen that magnifications of the orbit diagram tend to look "the same." In this experiment, we will see that there really is some truth to this: we will see that these period doubling bifurcations always occur at the same rate.
		This is the remarkable and surprising discovery made by the physicist Feigenbaum in 1975.
		The fact that these numbers are always the same is a remarkable result due to Feigenbaum several years ago.
	math.bu.edu /people/bob/MA471/lab4.html (472 words)

	Fractal Geometry (Site not responding. Last check: 2007-10-21)
		The speed at which Newton's method converges depends in part on how close the initial guess is to the true solution, and since Feigenbaum's pocket calculator was not especially fast, he began looking for patterns to produce better initial guesses.
		The limit of these ratios is the Feigenbaum delta constant,
		Feigenbaum's model for understanding this process relied upon the quadratic nature of the logistic map, so he was discouraged by the Metropolis, Stein, and Stein result that many other functions exhibit the same qualitative bifurcation behavior.
	classes.yale.edu /fractals/Chaos/LogUniv/UnivRatio.html (183 words)

	Feigenbaum’s constant
		A universal constant, denoted by the Greek letter, that governs the behavior of systems that are approaching chaos; it was discovered by the American mathematical physicist Mitchell Feigenbaum (1944-) in 1975 and has the value delta = 4.6692....
		All one-dimensional chaotic systems have a behavior, as they approach instability, known as period doubling.
		The Feigenbaum constant gives the rate at which the period of the system doubles.
	www.daviddarling.info /encyclopedia/F/Feigenbaums_constant.html (143 words)

	Notable Properties of Specific Numbers at MROB
		Khintchine's constant, the average value of the terms of the continued fraction for most (but not all) real numbers.
		The most well-known non-integral mathematical constant, the subject of several books, etc. Pi shows up in many places you don't expect it to (for instance, see 1.644934...
		In the Mandelbrot set, it shows up as the ratio between each "circle" and the next smaller one in the series of "circles" on the real axis connected to the large cardioid.
	home.earthlink.net /~mrob/pub/math/numbers-2.html (2000 words)

	Constants, Mu-Ency at MROB (Site not responding. Last check: 2007-10-21)
		This constant does not show up physically in the shape of the Mandelbrot Set, but there are various golden ratio places that you can find in the Mandelbrot Set's filament structure.
		This constant does not show up physically in the shape of the Mandelbrot Set, but it has been pointed out that e is involved in any work with complex numbers because (for example) e
		That's sort of a trivial relation, it's like saying that all circles are related to the number 7 because either a circle's radius or its area (or both) contain a 7 somewhere in their decimal expansion.
	www.mrob.com /pub/muency/constants.html (316 words)

	Welcome to Mathsoft
		Briggs, A precise calculation of the Feigenbaum constants, Math.
		J.-P. Eckmann and P. Wittwer, Computer Methods and Borel Summability applied to Feigenbaum's equation, Lecture Notes in Physics 227, Springer-Verlag, 1985; MR 86m:58129.
		Lanford, A computer-assisted proof of the Feigenbaum conjectures, Bull.
	www.mathsoft.com /mathsoft_resources/mathsoft_constants/ref/2091.asp (517 words)

	sci.fractals FAQ (Site not responding. Last check: 2007-10-21)
		Simply connected: X is simply connected if it is connected and every closed curve in X can be deformed in X to some constant closed curve.
		The equation is x -> c x (1 - x), where x is the population (between 0 and 1) and c is a growth constant.
		There is also an appendix giving the coordinates and constants for the color plates and many of the other pictures.
	www.faqs.org /faqs/sci/fractals-faq (12472 words)

	NUMBERS, MAGICK & MOTION III
		Here is a report out of Stanford University that shows a precise value of the fine-structure constant to 7.4 ppb....
		The result for the h/m cesium experiment for determination of the fine-structure constant is graphed exactly as: 137.0360005 + -- 7.4ppb....such that the equivalent using the ancient number 82944 is:
		The integer 37 also cracks the ratio to the Feigenbaum constant, ruler of the mandelbrot fractal, chaos to order phase transitions...F=4.669201609 = Feigenbaum constant...tangent in radians
	www.gnostics.com /numbersIII.html (1560 words)

	Rubidium (37) & Lanthanum (57) (Site not responding. Last check: 2007-10-21)
		Just as the blue diagonals in the triple dead center cube form a unit uniting the poles of the triple cube, so the atoms occupying logically blue places in the Rubidium crystal structure do likewise.
		If the total number of logical diagonals in the triple-Rubidium-3-cube crystal structure is 61 = 37+24, then, where 82944 is the common coefficient of the four fundamental force constants,
		On July 11, 2000, he communicated to the writer his observation concerning the numbers 37 and 57 as they appear in the periodic table.
	www.dgleahy.com /dgl/p23.html (1140 words)

	Feigenbaum Constant, Mu-Ency at MROB (Site not responding. Last check: 2007-10-21)
		Keith Briggs has computed the value of the constant to very high precision; here are the first 100 decimal places:
		It is possible that the first Feigenbaum constant can be used to directly compute the position of the Mandelbrot Set's center of gravity.
		Value of constant: Keith Briggs, Department of Applied Mathematics, University of Adelaide, South Australia 5005.
	www.mrob.com /pub/muency/feigenbaumconstant.html (328 words)

	[No title]
		When run, it will perform 9 iterations of a procedure designed to calculate an increasingly refined value of Feigenbaum's constant, starting from a rough initial approximation (3.2).
		For comparison purposes, the value of Feigenbaum's constant is 4.66920 16091029906718+.
		I ported Valentin's Feigenbaum program to the HP49G+, and it runs a little faster (as it should, compared to the 71B), i.e., 9 iterations in 1 min 35 sec.
	www.hpmuseum.org /cgi-sys/cgiwrap/hpmuseum/archv014.cgi?read=58017 (1442 words)

	Feigenbaums Constant (Site not responding. Last check: 2007-10-21)
		The fixpoints are very easy to find by simply repeating the iteration formula until the fixpoint cycle is reached.
		Feigenbaum's constant is defined to be the limit of
		You can get more information about Feigenbaum's tree here.
	www.math.carleton.ca /~amingare/chaosday/feigenbaum.html (80 words)

	Patterns in Chaos: The Feigenbaum Discovery
		This was the state of affairs in which Mitchell Feigenbaum, then of the Los Alamos National Laboratory, found himself while studying some very simple classes of functions in a way that no one had before.
		It led to the uncovering of some unexpected and universal patterns involving very simple functions.
		Explore the numbers and plots that underlie the Feigenbaum constant, and along the way find explanations of ideas central to chaos theory.
	www.wolfram.com /products/explorer/topics/chaos.html (109 words)

	Period doubling and Feigenbaum's scaling (Site not responding. Last check: 2007-10-21)
		You'll see the M-set self-similarity near the Feigenbaum point when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time.
		The same Feigenbaum's scaling symmetry with period doubling presents near every "mu-atom" (fl circle).
		You can also see amazing similarity between the Mandelbrot set filamentation ("antenna") near the cross position and the Julia set ("aeroplane").
	www.ibiblio.org /e-notes/MSet/Anim/Feigenbaum.htm (150 words)

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