Lorenz attractor

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The Lorenz attractor 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.

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 arises in lasers, dynamos, and specific waterwheels [1].

The equations that govern the Lorenz attractor are:

<math>\frac{dx}{dt} = \sigma (y - x)</math>

<math>\frac{dy}{dt} = x (\rho - z) - y</math>

<math>\frac{dz}{dt} = xy - \beta z</math>

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

[edit] The butterfly effect in the Lorenz attractor

Butterfly effect
Time t=1 (larger)	Time t=2 (larger)	Time t=3 (larger)
Image:Lorenz caos1-175.png	Image:Lorenz caos2-175.png	Image:Lorenz caos3-175.png
These figures — made using ρ=28, σ = 10 and β = 8/3 — show three time segments of the 3-D evolution of 2 trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10-5 in the x-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.
A Java animation of the Lorenz attractor shows the continuous evolution.

[edit] Using different values for the Rayleigh number

The Lorenz attractor for different values of ρ
Image:Lorenz Ro14 20 41 20-200px.png	Image:Lorenz Ro13-200px.png
ρ=14, σ=10, β=8/3 (larger)	ρ=13, σ=10, β=8/3 (larger)
Image:Lorenz Ro15-200px.png	Image:Lorenz Ro28-200px.png
ρ=15, σ=10, β=8/3 (larger)	ρ=28, σ=10, β=8/3 (larger)
For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.28, the fixed points become repulsors and the trajectory is repelled by them in a very complex way, evolving without ever crossing itself.
Java animation showing evolution for different values of ρ

[edit] See also

• List of chaotic maps
• Takens' theorem
• Mandelbrot

[edit] References

• Lorenz, E. N. (1963). "Deterministic nonperiodic flow". J. Atmos. Sci. 20: 130-141. doi:10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2.
• Frøyland, J., Alfsen, K. H. (1984). "Lyapunov-exponent spectra for the Lorenz model". Phys. Rev. A 29: 2928–2931.
• Tucker, W. (2002). "A Rigorous ODE Solver and Smale's 14th Problem". Found. Comp. Math. 2: 53-117.
• Strogatz, Steven H. (1994). Nonlinear Systems and Chaos. Perseus publishing. 
• Jonas Bergman, Knots in the Lorentz system, Undergraduate thesis, Uppsala University 2004.
• P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica D 9: 189-208. doi:10.1016/0167-2789(83)90298-1.

[edit] External links

• Eric W. Weisstein, Lorenz Attractor at MathWorld.
• Lorenz Equation on planetmath.org
• Lorenz Attractor Interactive Animation (you need the Adobe Shockwave plugin)
• Levitated.net: computational art and design
• 3D VRML Lorenz Attractor (you need a VRML viewer plugin)
• JAVA Applet - butterfly effect, Lorenz and Rossler attractors
• Essay on Lorenz Attractors in J - see J programming languagebe-x-old:Атрактар Лорэнца

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