Hopf bifurcation with cubic symmetry and instability of ABC

Abstract

We examine the dynamics of generic Hopf bifurcation in a system that is symmetric under the action of the rotational symmetries of the cube. We classify the generic branches of periodic solutions at bifurcation; there are generically 27 branches corresponding to maximal symmetries, organized into five symmetry types. There are also up to 22 periodic solution branches of two other symmetry types. These results are found by examination of the normal form (with S1 normal-form symmetry) for the bifurcation truncated at the third order. In addition to the periodic branches whose branching and stability we find, there are several branches of tori, homoclinic bifurcations and chaotic attractors in the dynamics of the third-order normal form. Since many of these features are not amenable to analysis, we give some numerical examples. On breaking the normal form symmetry, there may be breakup of the branches of tori, but the predictions for the periodic solutions will be reliable. For the Navier-Stokes equations with a particular forcing, an ABC flow is a dynamically stable solution for small Reynolds numbers R. For the most symmetric case, A = B = C = 1, the first instability of this system is a Hopf bifurcation at R = 13.04 with rotational symmetry of the cube. We use our normal-form analysis to explain the observed behaviour of solutions at this primary instability. Numerical simulations show that there is supercritical branching to rotating waves that alternate between the three axes, which undergo secondary Hopf bifurcation to a 2-torus at approximately R = 13.09. The eight symmetrically related tori break up and then merge to form a chaotic attractor with full symmetry. We can explain all these features by use of the generic third-order normal form and S1 normal-form symmetry-breaking terms.