| Abstract: | The present paper extends the singularity induced bifurcation theorem (SIBT) to higher index differential-algebraic equations (DAEs) in Hessenberg form. The SIBT arises in power system theory, and is also significant within the context of electrical circuits. This phenomenon is due to the presence of singularities in parameter-dependent problems, and it was originally proved for semiexplicit index-1 DAEs. We introduce a new proof of a matrix pencil-based version of this result, relying on the spectral features of the linearization of the underlying ordinary differential equation. This approach is then applied to higher index DAEs in Hessenberg form. For these structures, the SIBT is shown to follow from a minimal index change at the singularity. Additionally, we show that a Kronecker index change is also necessary for the existence of a singularity induced bifurcation point in semi-explicit index-1 and Hessenberg DAEs. |
