Lorenz Attractor

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 chaos 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. The system arises in lasers, dynamos and specific waterwheels.
\frac{dx}{dt} = \sigma (y - x)
\frac{dy}{dt} = x (r - z) - y
\frac{dz}{dt} = xy - b z
where \sigma is called the Prandtl number and r is called the Reynolds number. \sigma,r,b>0, but usually \sigma=10, b=8/3 and r is varied. The system exhibits chaos for r = 28, but displays knotted periodic orbits for other values of r, ie for r = 99.96 it becomes a T(3,2) torus knot.

References

  • Steven H. Strogatz, Nonlinear Systems and Chaos, Perseus publishing 1994.
  • Jonas Bergman, Knots in the Lorentz system, Undergraduate thesis, Uppsala University 2004.

External links

 

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