Bifurcation for second order Hamiltonian systems with periodic boundary conditions

Through variational methods, we study non--autonomous systems of second order ordinary differential equations with periodic boundary conditions. First we deal with a nonlinear system, depending on a function $u$, and prove that the set of bifurcation points for the solutions of the system is not $\sigma$--compact. Then we deal with a linear system depending on a real parameter $\lambda>0$ and on a function $u$, and prove that there exists $\lambda^*$ such that the set of the functions $u$, such that the system admits nontrivial solutions, contains an accumulation point.