Singularity-induced bifurcations in lumped circuits

A systematic analysis of singular bifurcations in semistate or differential-algebraic models of electrical circuits is presented in this paper. The singularity-induced bifurcation (SIB) theorem describes the divergence of one eigenvalue through infinity when an operating point or equilibrium locus of a parameterized differential-algebraic model crosses a singular manifold. The present paper extends this result to cover situations in which several eigenvalues diverge; we prove a multiple SIB theorem which states that a minimal rank (resp. index) change makes it possible to compute the number of diverging eigenvalues in terms of an index (resp. rank) change in the matrix pencil characterizing the linearized problem. The scope of the work comprises quasi-linear ordinary differential equations, semiexplicit index-1 differential-algebraic equation (DAEs), and Hessenberg index-2 DAEs, describing different electrical configurations. The electrical features from which singularities and, specifically, singular bifurcations stem are extensively discussed. Examples displaying simple, double, and triple SIB points illustrate different ways in which the spectrum may diverge. Index Terms--Bifurcation, differential-algebraic equation (DAE), index, matrix pencil, nonlinear circuit, semistate equation, singularity.