The twisting structure seen in the image below represents an invariant torus in a three dimensional fluid flow. The flow of interest is known as the Kuppers-Lortz instability which arises in a rapidly rotating thin fluid layer heated from below (Rayleigh-Benard convection). If the temperature difference exceeds a critical value, convection cells form in various patterns.

front/pocket

One example is rolls which are a stable solution of the Boussinesq equations. However, if the rotation rate of the layer exceeds a threshold value, then the rolls are unstable to rolls rotated 120 degrees with respect to the original. These rolls are also unstable to another set at 240 degrees. The 240 degree rolls are unstable to the original set at 0 degrees and the pattern continues indefinitely.

This behavior can be described simply by a set of nonautonomous ordinary differential equations. The structure is interpreted as tracer particle trajectories (like a dye in a fluid) following a complicated toroidal structure. The torus is embedded in a chaotic sea where the mixing of dye is thorough. However, inside the torus the behavior is uniform and simple (less complicated) where no mixing occurs.