- Lorenz attractor
The Lorenz attractor, named for
Edward N. Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its butterflyshape. The map shows how the state of a dynamical system(the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.
The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in
1963, who derived it from the simplified equations of convectionrolls arising in the equations of the atmosphere.
From a technical standpoint, the system is nonlinear, three-dimensional and deterministic. In 2001 it was proven by Warwick Tucker that for a certain set of parameters the system exhibits chaotic behavior and displays what is today called a strange attractor. The strange attractor in this case is a
fractalof Hausdorff dimensionbetween 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimensionto be 2.05 ± 0.01.
The system also arises in simplified models for
lasers harv|Haken|1975 and dynamos harv|Knobloch|1981.
The equations that govern the Lorenz attractor are::
where is called the
Prandtl numberand is called the Rayleigh number. All , , > 0, but usually = 10, = 8/3 and is varied. The system exhibits chaotic behavior for = 28 but displays knotted periodic orbits for other values of . For example, with it becomes a "T"(3,2) torus knot.
The source code to simulate the Lorenz attractor in
## Lorenz Attractor equations solved by ODE Solve
## x' = sigma*(y-x)
## y' = x*(rho - z) - y
## z' = x*y - beta*zfunction dx = lorenzatt(X,T) rho = 28; sigma = 10; beta = 8/3; dx = zeros(3,1); dx(1) = sigma*(X(2) - X(1)); dx(2) = X(1)*(rho - X(3)) - X(2); dx(3) = X(1)*X(2) - beta*X(3); returnend
## Using LSODE to solve the ODE system.clear allclose alllsode_options("absolute tolerance",1e-3)lsode_options("relative tolerance",1e-4)t = linspace(0,25,1e3); X0 = [0,1,1.05] ; [X,T,MSG] =lsode(@lorenzatt,X0,t);TMSGplot3(X(:,1),X(:,2),X(:,3))view(45,45)
List of chaotic maps
* Jonas Bergman, "Knots in the Lorentz system", Undergraduate thesis, Uppsala University 2004.
* cite journal
author=P. Grassberger and I. Procaccia
title=Measuring the strangeness of strange attractors
year = 1983 | volume = 9 | pages=189–208
url = http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983PhyD....9..189G&db_key=PHY
doi = 10.1016/0167-2789(83)90298-1
* [http://demonstrations.wolfram.com/LorenzAttractor/ Lorenz attractor] by Rob Morris,
The Wolfram Demonstrations Project.
* [http://planetmath.org/encyclopedia/LorenzEquation.html Lorenz equation] on planetmath.org
* [http://www.mizuno.org/c/la/index.shtml For drawing the Lorenz attractor, or coping with a similar situation] using
ANSI Cand gnuplot.
* [http://video.google.com/videoplay?docid=2875296564158834562&q=strogatz&ei=xr9OSJ_SOpeG2wKB3Iy2DA&hl=en Synchronized Chaos and Private Communications, with Kevin Cuomo] . The implementation of Lorenz attractor in an electronic circuit.
* [http://toxi.co.uk/lorenz/ Lorenz attractor interactive animation] (you need the Adobe Shockwave plugin)
* [http://www.levitated.net/daily/levLorenzAttractor.html Levitated.net: computational art and design]
* [http://ibiblio.org/e-notes/VRML/Lorenz/Lorenz.htm 3D VRML Lorenz attractor] (you need a VRML viewer plugin)
* [http://www.jsoftware.com/jwiki/Essays/Lorenz_Attractor Essay on Lorenz attractors in J] - see
J programming language
Wikimedia Foundation. 2010.
Look at other dictionaries:
Lorenz attractor — … Useful english dictionary
Lorenz — may refer to:Law Firm* Lorenz International Lawyers, an international law firm with offices in Brussels, Bishkek and GenevaMusic* Christian Flake Lorenz, a German musician * Lorenz Hart, the lyricist half of the famed Broadway songwriting team… … Wikipedia
Attractor — For other uses, see Attractor (disambiguation). Visual representation of a strange attractor An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if… … Wikipedia
Lorenz , Edward Norton — (1917–) American meteorologist Born in West Hartford, Connecticut, Lorenz was educated at Dartmouth College, New Hampshire, at Harvard, and at the Massachusetts Institute of Technology. He joined the MIT faculty in 1946 and served as professor of … Scientists
Lorenz-Attraktor — Grafische Darstellung eines Lorenz Attraktors Der Lorenz Attraktor ist der seltsame Attraktor eines Systems von drei gekoppelten, nichtlinearen gewöhnlichen Differentialgleichungen: F … Deutsch Wikipedia
Rössler attractor — The Rössler attractor (pronEng|ˈrɒslɚFact|date=December 2007) is the attractor for the Rössler system, a system of three non linear ordinary differential equations. These differential equations define a continuous time dynamical system that… … Wikipedia
Atractor de Lorenz — Saltar a navegación, búsqueda El atractor de Lorenz, con valores r = 28, σ = 10, b = 8/3 … Wikipedia Español
Edward Norton Lorenz — Infobox Scientist box width = 300px name = Edward Norton Lorenz image width = 200px caption = Edward Norton Lorenz birth date = birth date|1917|5|23 birth place = West Hartford, Connecticut, United States death date = death date and… … Wikipedia
Attracteur de Lorenz — Pour consulter un article plus général, voir : théorie du chaos. L attracteur de Lorenz En 1963, le météorologue Edward Lorenz est le premier à mettre en évidence le caractère vraisemblablement chaotique de la météorologie. Le modèle de… … Wikipédia en Français
Chaos theory — This article is about chaos theory in Mathematics. For other uses of Chaos theory, see Chaos Theory (disambiguation). For other uses of Chaos, see Chaos (disambiguation). A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 … Wikipedia