  The index of a selfadjoint Fredholm operator is zero by the well-known fact
that the kernel of a selfadjoint operator is perpendicular to its range. The
Fredholm index was generalised to families by Atiyah and J\"anich in the
sixties, and it is readily seen that on complex Hilbert spaces this so called
index bundle vanishes for families of selfadjoint Fredholm operators as in the
case of a single operator. The first aim of this note is to point out that for
every real Hilbert space and every compact topological space $X$ there is a
family of selfadjoint Fredholm operators parametrised by $X\times S^1$ which
has a non-trivial index bundle. Further, we use this observation and a family
index theorem of Pejsachowicz to study multiparameter bifurcation of homoclinic
solutions of Hamiltonian systems, where we generalise a previously known class
of examples.
