The Singularity-Induced Bifurcation and its Kronecker Normal Form

Abstract

It is shown that the singularity-induced bifurcation theorem due to Venkatasubramanian, Schattler, and Zaborszky [ Proceedings of the IEEE, 83 (1995), pp. 1530--1558] can be expressed as the perturbation of an infinite eigenvalue of a particular class of parameterized index-1 matrix pencil, denoted $(M,L(\lambda))$. It is shown that the matrix pencil at the singularity-induced bifurcation point, $(M,L(\lambda_{0}))$, has Kronecker index-2. It is also shown that a two-parameter unfolding of a singularity-induced bifurcation point results in a locus of index-0 pencils, denoted $(M(\epsilon),L(\lambda(\epsilon)))$, which has two purely imaginary eigenvalues near infinity.