Detection of Tertiary Hopf Bifurcations Arising from Mode Interactions

Abstract

In the unfolding of a mode interaction, in addition to the primary bifurcations, there are also secondary bifurcations which occur on the primary branches giving rise to mixed mode solutions. A further tertiary Hopf bifurcation arises in some cases from the mixed mode solutions. The detection of Hopf bifurcation points is a numerically expensive procedure and so we consider whether it is possible to predict the existence of the tertiary Hopf bifurcation by considering only the geometric structure of the primary and secondary branches. We show that in some cases, it is possible to show that no Hopf bifurcation exists while in other cases, more information in the form of the stability of the trivial solution is required to determine whether or not the Hopf bifurcation exists. An algorithm for determining the existence of the Hopf bifurcation is given.