Behavior of a Three-Torus in Truncated Navier-Stokes Equations

The presence of a three-torus in a seven-mode truncation of the three-dimensional Navier-Stokes equations is investigated numerically by means of cross-section and power spectra. Furthermore, by taking advantage of particular features of the model, rotation vectors, circle maps and torus maps can be computed with high accuracy and used to study the dynamics. In particular, some interesting phenomena of partial phase-locking are described in deep detail. The three-torus, which arises via a Hopf bifurcation and persists in a wide parameter range, is found to break and originate a strange attractor. The onset of chaos and the associated bifurcation point can be defined quite precisely.