Hopf Bifurcation and Stability in DAEs

A simple comparison of the areas of the sciences in which DAEs are involved with those in which examples of Hopf bifurcation in ODEs arise (see Thompson and Stewart (1986) and the references therein) reveals a considerable overlap and suggests that an appropriate variant of the Hopf bifurcation theorem should be available in the DAE setting. In this chapter, we show that, indeed, a generalization of the Hopf bifurcation theorem to DAEs can be phrased without any reference to a reduction (or augmentation). In particular, the conditions for Hopf bifurcation are the same irrespective of the index of the DAE. This is a rather pleasant surprise since, as a rule, everything else with DAEs becomes more intricate as the index increases.