- 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
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