Spectral flow and bifurcation of critical points of strongly indefinite functionals. II. Bifurcation of periodic orbits of Hamiltonian systems.

Our main results here are as follows: Let X* be a family of 2?-periodic Hamiltonian vectorfields that depend smoothly on a real parameter * in [a, b] and has a known, trivial, branch s* of 2?-periodic solutions. Let P* be the Poincare map of the linearization of X* at s* . If the ConleyZehnder index of the path P* does not vanish, then any neighborhood of the trivial branch of periodic solutions contains 2?-periodic solutions not on the branch. Moreover, if each solution s* is constant and each linearization A* of X* at s* is time independent, then bifurcation of 2?-periodic orbits from the branch of equilibria arises whenever i(Ab){i(Ab), where i(A) is the index of the linear Hamiltonian system Ju*=Au.