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 $\lambda $. 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 $\lambda $ 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.