The Lorenz Attractor

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

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

\(\frac {dz}{dt} = -\beta z + x y \)

The parameter values that Lorenz used in the paper are \(\sigma = 10\), \(\rho = 28\), and \(\beta = 8/3\). All pictures of the Lorenz atractor in this web page use these parameters.

One can think of a particle moving in 3-D space using the rule that its velocity vector, \( \frac {d} {dt} ( x, y, z )\), is a given function of its position, \( (x,y,z) \). (This is like Aristotelian physics which says that gravity makes objects move with a downward velocity.) A solution to the Lorenz equations is a vector valued funtion \(t \mapsto \big (x(t), y(t), z(t) \big )\) such that \(x'(t) = \sigma \big (y(t) - x(t)\big )\) for all \(t\), with a similar translation of the second and third equation. We cannot get closed-form solutions but need to solve the ODE numerically.
The first figure shows a short-time approximation of the two braches of the unstable manifold, one in red and one in blue. If you click on the figure, you will see an animation (531 K) where the figure rotates. To make this animate gif smaller, the figure only rotates 180 degrees. This has the effect of switching the colors instantaneoudly every time we loop back to the beginning.

If we let time grow without bound, the unstable manifold would fill in the spaces between the curves, but not fill in the two ``eyes". The closure of the unstable manifold is the Lorenz attractor. It is approximated by a branched manifold.

This second figure reveals the branching of the attractor. The figure plots periodic orbits of type LR (black), LLRR (green), LLLR (red), and RRRL (blue). If you click on this figure, you will see an animation (442 K) where the attractor tilts down. These periodic orbits are all unstable. In fact, unstable periodic orbits are dense in the Lorenz attractor, but there are no stable periodic orbits.

Poincare Map for the Sequence of L's and Rs

Finding periodic orbits in the Lorenz Attractor