Spectral Flow and Bifurcation of Critical Points of Strongly Indefinite Functionals: Part II. Bifurcation of Periodic Orbits of Hamiltonian Systems

Abstract

Our main results here are as follows: Let Xλ be a family of 2π-periodic Hamiltonian vectorfields that depend smoothly on a real parameter λ in [a, b] and has a known, trivial, branch sλ of 2π-periodic solutions. Let Pλ be the Poincaré map of the linearization of Xλ at sλ. If the Conley–Zehnder index of the path Pλ does not vanish, then any neighborhood of the trivial branch of periodic solutions contains 2π-periodic solutions not on the branch. Moreover, if each solution sλ is constant and each linearization Aλ of Xλ at sλ is time independent, then bifurcation of 2π-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.