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.

A plot of the Lorentz-system for specific values of the parameters

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

$\frac{dy}{dt} = x (r - z) - y$

$\frac{dz}{dt} = xy - b z$

where $σ$ is called the Prandtl number and r is called the Reynolds number. $σ, r, b > 0$ , but usually $σ = 10$ , $b = 8 / 3$ 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

Categories: Dynamical systems