Modelling instability of 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 = 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 = 13.044 , 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. neric 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 = 13.91 . This includes a torus breakdown to a chaotic attractor and a crisis of attractors that leads to a change in symmetry. cuss numerical simulations of the hydrodynamic system for larger values of R . 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 = 15 .