The Lorenz Attractor

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

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

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

The standard parameter values which Lorenz used in the paper are \(\sigma = 10\), \(r = 28\), and \(b = 8/3\).

These equations are derived by Saltzman 1962 and Lorenz 1963 for thermal convention in a fluid layer heated from below. In those papers, the variables \(x\), \(y\) and \(z\) denote the usual 3-D space we live in. The variables in the Lorenz paper are \(X\), \(Y\) and \(Z\) (capital letters). They are three coefficients in the Fourier series for the stream function \(\psi (x,z, t) = X(t) \sin(\pi x/\sqrt{2}) \sin(\pi z) + \cdots\) and the temperature distribution \(T(x,z, t) = 1 - z + Y(t) \cos(\pi x/\sqrt{2}) \sin(\pi z) - Z(t)/(2 \pi) \sin(2 \pi z) + \cdots\). These figures showing the stream function and temperatrue distribution were created with this Mathematica notebook. The coefficients (\(X, Y, Z)\) are just the first three of an infinite set. Saltzman had previously truncated the system to 7 Fourier coefficients, which he called (\(A,B,\ldots, G)\). These truncations are valid for \(\rho = 1 + \varepsilon\), showing the transition from conduction (\(X = Y = Z = 0\)) to small but finite amplitude convection for which \(X = Y \neq 0\) and \(Z > 0\) are constant functions of \(t\).

Now that the Lorenz equations are derived, we get rid of the capitol letters. One can think of the Lorenz equations as describing the motion of a particle moving in 3-D space with position \(( x(t), y(t), z(t)) \), using the rule that its velocity vector \( \frac {d} {dt} ( x, y, z )\) is a given function of its position. (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, for the standard parameter values. 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

Here is a Mathematica notebook LorenzPoincareMapUnstableManifold.nb that estimates that the unstable manifold spins around the ``eye'' 25 times.

Here is a Mathematica notebook LorenzMaxZ.nb that reproduces the one-dimensional map from the Lorenz63 paper. The map has very few points near the cusp.

March 3: Here is the filled in LorenzMap.pdf, which was produced by this Mathematica Notebook: Lorenz1Dmap.nb

March 3: Here is the LorenzSnail.pdf, which was produced by this Mathematica Notebook: LorenzSnailp.nb. It can be used to make a model of the "Geometric Lorenz attractor" that lies on a branched manifold.
Here is a pdf, LorenzStableAndUnstable.pdf, of the single trajectory that is used to make the snail. The trajectory shows the start of both the stable and unstable manifold of the origin.

Here is a Mathematica notebook LorenzFindStableManifold.nb that finds a point in the attractor that goes to the origin as time goes to infinity.

Here is a Desmos 3D file that plots the Lorenz flow near (0,0,0). These are solutions to the linearized ODE near (0,0,0), in the y coordinates (for which the 3 ODEs are decoupled). Hint: Click settings, adjust perspective and set translucent surfaces.

Here is a Desmos 3D file that plots the Lorenz anvil, which is the image of the red rectangle under the flow.