Lorenz attractor

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The Lorenz attractor, introduced by Edward Lorenz in 1963, is a non-linear three-dimensional deterministic dynamical system derived from the simplified equations of convection rolls arising in the dynamical equations of the atmosphere. In 2001 it was proven by W. 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].

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

where <math>\sigma</math> is called the Prandtl number and <math>\rho</math> is called the Rayleigh number. <math>\sigma, \rho, \beta >0</math>, but usually <math>\sigma = 10</math>, <math>\beta = 8/3</math> and <math>\rho</math> is varied. The system exhibits chaotic behavior for <math>\rho = 28</math> 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.

The butterfly-like shape of the Lorenz attractor may have inspired the name of the butterfly effect in chaos theory.

1 The butterfly effect in the Lorenz attractor
2 Using different values for the Rayleigh number
3 See also
4 References
5 External links

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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.

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.

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Using different values for the Rayleigh number

The Lorentz 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 atractors. When ρ is larger than 24.74, 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

The Lorentz 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 atractors. When ρ is larger than 24.74, 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

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References

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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
| id = Template:Doi
}}

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