Splitting of Steady Multiple Eigenvalues May Lead to Periodic Cascading Bifurcation

A general bifurcation problem is considered that depends on two parameters in addition to the bifurcation parameter λ. It is assumed that all primary bifurcation states correspond to steady solutions and that they branch supercritically. Then it is shown that for a range of system parameters and near a triple primary bifurcation point the following cascade of bifurcations from the minimum primary bifurcation state is possible. As λ increases there is secondary and then tertiary bifurcation to steady states and finally Hopf bifurcation at a quarternary bifurcation point. Related transitions have been observed experimentally in thermal convection and other hydrodynamic stability problems. In addition, we show that Hopf bifurcation near a double primary bifurcation point is not possible when both primary states near the double point bifurcate supercritically. However, it is possible near such a double bifurcation point if imperfections are included in the formulation, as we demonstrate.