Bifurcation theory and the type numbers of marston morse

Let H be a real Hilbert space and f(x,lambda) be a C(2) operator mapping a small neighborhood U of (x(0),lambda(0)) epsilon (H x R(1)) into itself. We investigate the solutions of the equation f(x,lambda) = 0 near a solution (x(0),lambda(0)), assuming that f(x,lambda) is a gradient mapping and 0 < dim Ker f(x)(x(0),lambda(0)) < infinity. In particular, we show that the type numbers of Marston Morse for an isolated critical point can be used to prove the existence of a point of bifurcation at (x(0),lambda(0)). An application of this result is given to the discovery of periodic motions near a stationary point for a large class of nonlinear Hamiltonian systems in "resonant" cases.