Modelling instability of ABC flow using a mode interaction between steady and Hopf bifurcations with rotational symmetries of the cube

We consider hydrodynamic stability of symmetric ABC flow with A=B=C=1A=B=C=1, regarded as a steady state of the three-dimensional Navier–Stokes equation with appropriate forcing. Numerical investigations have shown that its first instability on increasing the Reynolds number is a Hopf bifurcation at R=13.044R=13.044, and that at this bifurcation the second dominant eigenvalue is real. Motivated by this, we study generic interaction of steady-state and Hopf bifurcations in systems with rotational symmetry of the cube O with an eight-dimensional normal form.The generic branching and bifurcation behavior of the third-order truncated normal form is investigated. This is used to analyse a range of the bifurcations for particular values of the coefficients, obtained by center manifold reduction from the hydrodynamic system. The normal form system shows a sequence of bifurcations and attractors that closely follows the sequence observed for the original hydrodynamic system up to about R=13.91R=13.91. This includes a torus breakdown to a chaotic attractor and a crisis of attractors that leads to a change in symmetry.We discuss numerical simulations of the hydrodynamic system for larger values of RR. Finally, we present evidence that the system has robust heteroclinic cycles between fully symmetric ABC flow and six steady states with broken symmetry for a range of parameter values near R=15R=15.