Lorenz attractor

From Freepedia

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. For a certain set of parameters the system exhibits chaotic behavior and displays what is today called a strange attractor; this was proven by W. Tucker in 2001. 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 r is called the Reynolds number. <math>\sigma,r,b>0</math>, but usually <math>\sigma=10</math>, <math>b=8/3</math> and r is varied. The system exhibits chaotic behavior for r = 28 but displays knotted periodic orbits for other values of r. For example, with r = 99.96 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.

References

Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci. 20: 130-141.
Frøyland, J., Alfsen, K. H. (1984). Lyapunov-exponent spectra for the Lorenz model. Phys. Rev. A 29: 2928–2931.
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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. (LINK)