Double SIB points in differential-algebraic systems

A singularity-induced bifurcation (SIB) describes the divergence of one eigenvalue through infinity when an equilibrium locus of a parameterized differential-algebraic equation (DAE) crosses a singular manifold. The present note extends the analysis of this behavior to cover double SIB points, for which two eigenvalues diverge. The key assumption supporting this phenomenon is that the Kronecker index jumps by two at the singularity. In this situation, double SIB points are shown to undergo generically a transition from a spiral to a saddle in the linearized problem, after restricting the analysis to the corresponding invariant subspace. Typical examples arise in the context of nonlinear RLC circuits. The setting for the study is that of semi-explicit DAEs in Hessenberg form with arbitrary index.