The Lorenz attractor is a chaotic map, 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, often described as beautiful.

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 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 ^*.

The equations that govern the Lorenz attractor are:

$\backslash frac\{dx\}\{dt\}\; =\; \backslash sigma\; (y\; -\; x)$

$\backslash frac\{dy\}\{dt\}\; =\; x\; (\backslash rho\; -\; z)\; -\; y$

$\backslash frac\{dz\}\{dt\}\; =\; xy\; -\; \backslash beta\; z$

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

The butterfly effect in the Lorenz attractor

Butterfly effect
Time t=1 Lorenz_caos1.png	Time t=2 Lorenz_caos2.png	Time t=3 Lorenz_caos3.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.

Using different values for the Rayleigh number

The Lorenz attractor for different values of ρ
ρ=14, σ=10, β=8/3 Lorenz_Ro14_20_41_20.png	ρ=13, σ=10, β=8/3 Lorenz_Ro13.png
ρ=15, σ=10, β=8/3 Lorenz_Ro15.png	ρ=28, σ=10, β=8/3 Lorenz_Ro28.png
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

See also

• List of chaotic maps
• Takens' theorem

References

• Jonas Bergman, Knots in the Lorentz system, Undergraduate thesis, Uppsala University 2004.

External links

• Lorenz Attractor (MathWorld article)
• Lorenz Equation on planetmath.org
• 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 language

Chaotic maps

Атрактар Лорэнца | Lorenzův atraktor | Lorenz-Attraktor | Atractor de Lorenz | 로렌츠 방정식 | Attrattore di Lorenz | ローレンツ方程式 | Układ Lorenza | Аттрактор Лоренца | ตัวดึงดูดลอเรนซ์