Finite aspect ratio Taylor–Couette flow: Shil’nikov dynamics of 2-tori

The nonlinear dynamics of the flow in a short annular container driven by the rotation of the inner cylinder is studied using direct numerical simulations of the three-dimensional Navier–Stokes equations. The basic state is S O ( 2 ) × Z 2 symmetric. For aspect ratios between 3.6 and 4.4, we have located three codimension-two bifurcations: a cusp, a double Hopf and a fold-Hopf bifurcation. All these local bifurcations are Z 2 -invariant. The breaking of Z 2 symmetry involves very complex Shil’nikov-type dynamics, not directly connected to any of the three codimension-two bifurcations, but associated with five unstable limit cycles and a wealth of heteroclinic connections between them. Period-adding cascades, both direct and reverse, of 2-tori have been found. Narrow regions of chaotic dynamics are interspersed within these quasiperiodic solutions.