The singularity-induced bifurcation and its Kronecker normal form

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.