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.

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

<math>\frac{dy}{dt} = x (r – z) – y<math>

<math>\frac{dz}{dt} = xy – b z<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 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.

See also Takens' theorem.

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

• Lorentz Attractor
• Plot of the Lorenz attractor
• planetmath.org: Lorenz Equation
• Levitated.net: computational art and design