When the Morse index is infinite

We prove that if f is a functional on a Hilbert manifold M having critical points with infinite Morse index and co-index, the following fact holds: for every arbitrary choice of an integer a(x) for each critical point x, there exists a Riemannian metric on M such that the gradient flow of f is Morse-Smale and the intersection of the unstable manifold of x with the stable manifold of y has dimension a(x)-a(y). This fact shows that for strongly indefinite functionals, no Morse theory based only on the data (M,f) can exist