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 butterfly shape. 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.

Overview

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 convection rolls 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 fractal of Hausdorff dimension between 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 2.05 ± 0.01.

The system also arises in simplified models for lasers harv|Haken|1975 and dynamos harv|Knobloch|1981.

Equations

The equations that govern the Lorenz attractor are:: $frac\{dx\}\{dt\}\; =\; sigma\; (y\; -\; x)$

: $frac\{dy\}\{dt\}\; =\; x\; (\; ho\; -\; z)\; -\; y$

:

where $sigma$ is called the Prandtl number and $ho$ is called the Rayleigh number. All $sigma$, $ho$, > 0, but usually $sigma$ = 10, = 8/3 and $ho$ is varied. The system exhibits chaotic behavior for $ho$ = 28 but displays knotted periodic orbits for other values of $ho$. For example, with $ho\; =\; 99.96$ it becomes a "T"(3,2) torus knot.

Butterfly effect

Rayleigh number

ource Code

The source code to simulate the Lorenz attractor in GNU Octave follows.

## 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)

See also

* List of chaotic maps
* Takens' theorem
* Mandelbrot set

References

* 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
journal=Physica D
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
*.
*
*.
*
*

External links

*
* [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 C and 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