A family index theorem for periodic Hamiltonian systems and bifurcation

Abstract

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah–Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theorem to study bifurcation of branches of periodic solutions for families of nonlinear Hamiltonian systems.