On the singularity-induced bifurcation theorem

The singularity-induced bifurcation theorem (SIBT) is extended in this note to quasi-linear singular ordinary differential equations. The hypotheses supporting this result are simplified and rewritten in terms of matrix pencils. This approach shows that the SIB follows from a minimal index change at the singularity. The use of a quasi-linear reduction leads to a simple statement of the SIBT for semiexplicit index-1 differential algebraic equations.