  We consider families of strongly indefinite systems of elliptic PDE and
investigate bifurcation from a trivial branch of solutions by using the
spectral flow. The novelty in our approach is a refined version of a comparison
principle that was originally proved by Pejsachowicz in a joint work with the
third author, and which is based on an index theorem for a certain class of
Fredholm operators that is of independent interest. Finally, we use our
findings for a bifurcation problem on shrinking domains that originates from
works of Morse and Smale.
