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# Topic: Lyapunov fractal

 Fractal - Wikipedia, the free encyclopedia The term fractal was coined in 1975 by BenoĆ®t Mandelbrot, from the Latin fractus or "broken". Fractal geometry is the branch of mathematics which studies the properties and behaviour of fractals. Fractal geometry was also used for data compression and for modelling complex organic and geological systems, for example the growth of trees or the development of river basins. en.wikipedia.org /wiki/Fractal   (1929 words)

 Aleksandr Lyapunov - FreeEncyclopedia   (Site not responding. Last check: 2007-11-06) Lyapunov had already begun to study this stability in his previous two-years attempts at solving the task. Lyapunov had lectured already from 1880 at the faculty for mechanics and this had taken him a lot of time. Lyapunov lectured at the university on themes from theoretical mechanics, integrals of differential equations and the theory of probability. openproxy.ath.cx /al/Aleksandr_Mikhailovich_Lyapunov.html   (1609 words)

 Fractal - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-06) Fractals are generally irregular (not smooth) in shape, and thus are not objects definable by traditional geometry. Some common examples of fractals include the Mandelbrot set, Lyapunov fractal, Cantor set, Sierpinski gasket and carpet, Menger sponge, dragon curve, Peano curve, and the Koch curve. These naturally occurring fractals (like clouds, mountains, river networks, and systems of blood vessels) have both lower and upper cut-offs, but they are separated by several orders of magnitude. www.hackettstown.us /project/wikipedia/index.php/Fractal   (1922 words)

 Fractal - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-06) The term fractal is now used as a scientific concept, as well as a strictly mathematical idea. Fractals such as the Mandelbrot set are more loosely self-similar: they contain small copies of the entire fractal in distorted and degenerate forms. Fractal techniques have also been employed in image compression, as well as a variety of scientific disciplines. www.indexuslist.de /keyword/Fractal.php   (1130 words)

 Fractal   (Site not responding. Last check: 2007-11-06) In many cases a fractal can be generated (for example on a computer screen) by a repeating pattern, typically a recursive or iterative process. Fractals of many kinds were originally studied as mathematical objects, and the term "fractal" has been given various precise definitions by mathematicians. Fractal geometry rescues this situation by providing new tools which are able to accommodate this detail. www.yotor.com /wiki/en/fr/Fractal.htm   (1989 words)

 FRACTINT Lyapunov Fractals   (Site not responding. Last check: 2007-11-06) The Lyapunov fractal expands that model into two dimensions by letting the growth rate vary in a periodic fashion between two values. Negative Lyapunov exponents indicate a stable, periodic behavior and are plotted in color. When two parts of a Lyapunov overlap, which spike overlaps which is strongly dependent on the initial value of the population model. spanky.triumf.ca /www/fractint/lyapunov_type.html   (469 words)

 sci.fractals FAQ Fractals also describe many real-world objects, such as clouds, mountains, turbulence, coastlines, roots, branches of trees, blood vesels, and lungs of animals, that do not correspond to simple geometric shapes. Roughly, fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. Topics include: challenges of using fractals in the classroom, new ways of generating art and music, the use of fractals in clothing fashions of the future, fractal holograms, fractals in medicine, fractals in boardrooms of the future, fractals in chess. www.faqs.org /faqs/sci/fractals-faq   (12472 words)

 Fractal Frequently Asked Questions and Answers Roughly, fractal dimension can be calculated by taking the limit of the quo- tient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. Fractal dimensions quantify the static *geometry* of an object. The aim of this first international journal on fractals is to bring together the most recent developments in the research of fractals so that a fruitful interaction of the various approaches and scientific views on the complex spatial and temporal behavior could take place. www.faqs.org /faqs/fractal-faq   (9659 words)

 Fractal   (Site not responding. Last check: 2007-11-06) Harrison http://math.berkeley.edu/~harrison/research/publications/ extended Newtonian calculus to fractal domains, including the theorems of Gauss, Green, and Stokes. Image:Julia_set_(indigo).pngA Julia set, a fractal related to the Mandelbrot set Image:Glue1_800x600.jpgPulling two glue-covered acrylic sheets apart forms a natural fractal. Fractovia - authoritative source of fractal generators, as well as a listing of tutorials regarding fractals in general, and specific programs. www.worldhistory.com /wiki/F/Fractal.htm   (1855 words)

 Extracting beauty from chaos Like most fractal images, Lyapunov Exponent fractals are produced by iterating functions and observing the chaotic behaviour that may result. Alexander M. Lyapunov, 1857-1918) is used to measure the extent to which ever-smaller "infinitesimal" errors are amplified. The Lyapunov Exponent is defined to be the limiting value of the above quantity as the initial error E[0] is made ever smaller-and-smaller, and the number of iterations n is sent to infinity. plus.maths.org /issue9/features/lyapunov   (2542 words)

 sci.fractals FAQ For more information on fractal dimension and scale, via the WWW Fractals and Scale (by David G. Green) http://life.csu.edu.au/complex/tutorials/tutorial3.html Fractal dimension references: 1. Fractal generated clouds http://ivory.nosc.mil/html/trancv/html/cloud-fract.html The Center for the Computation and Visualization of Geometric Structures also has some fractal clouds. MS-DOS and Unix programs for estimating Lyapunov exponents from short time series are available by ftp: ftp://inls.ucsd.edu/pub/ncsu/ Computing Lyapunov exponents in general is more difficult. www.cs.uu.nl /wais/html/na-dir/sci/fractals-faq.html   (11671 words)

 Fractal Frequently Asked Questions and Answers FAQ There are many fractal programs available from oak.oakland.edu:/pub/msdos/graphics [141.210.10.117]: forb01a.zip: Displays orbits of Mandelbrot mapping. Fractal images including some recent alt.binaries.pictures.fractals images are archived at spanky.triumf.ca:/fractals [142.90.112.1]. A25: There are several Internet sites with fractal papers: There is an ftp archive site for preprints and programs on nonlinear dynamics and related subjects at: lyapunov.ucsd.edu:/pub [132.239.86.10]. www.non.com /news.answers/fractal-faq.html   (9552 words)

 Godric's Fractal Gallery   (Site not responding. Last check: 2007-11-06) Some fractals are extremely self-similar while others are hardly at all. The large image on the left is the entire fractal at magnification 1. This Markus-Lyapunov fractal is a relative of the bifurcation fractal (honest!) www.gozen.demon.co.uk /godric/fracgall.html   (878 words)

 Omniseek: /Science & Tech /Math /Mathematical Recreations /Fractals   (Site not responding. Last check: 2007-11-06) Fractals and Chaos look for Mandelbrot Set, Julia Set Tutorial about Mandelbrot and Julia sets. Lyapunov type (type=lyapunov) The Bifurcation fractal illustrates what happens in a simple population model as the growth rate increases. The Lyapunov fractal expands that model into two dimensions by... www.omniseek.com /srch/{48258}   (193 words)

 [No title] In fact there's no need to rotate the string at all, it may have leading b's; the length of the string is input in a separate variable (abMax). The conditions here at the new Fractal Central auxiliary are still a bit hectic, but they are becoming more organized every day. The fractal weather today was mostly cloudy and a chilly 50F (10C) -- conditions that combined with a brisk wind to keep the fractal cats indoors sulking most of the afternoon. www.xmission.com /pub/lists/fractint/archive/v01.n548   (3790 words)

 Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations Energy Citations Database (ECD) Document #5419401 - Estimating the fractal dimension and the predictability of the atmosphere Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link. Estimating the fractal dimension and the predictability of the atmosphere www.osti.gov /energycitations/product.biblio.jsp?osti_id=5419401   (114 words)

 [No title] 40v } event_h2 { ; par and image Copyright 1998, Paul Kyle ; Lyapunov fractal Paul.frac reset=1960 type=lyapunov center-mag=-9.09495e-012/-0.153548/0.0001667628 params=16/1.020202e-006/139 float=y maxiter=147 invert=6/4/0 decomp=8 colors=00000z I really like the way this one turned out - I hope you do also. I render all my images at 1024x768, and so I hacked= my fractint.cfg to remove all the extraneous entries except for the 1024x768 Vesa mode (f2), the 1024x768 disk video mode (sf2), and a few other lower res modes for previewing. www.xmission.com /pub/lists/fractint/archive/v01.n340   (1834 words)

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