On a System with Multiple Period Annuli

We consider two-point boundary value problems for the Hamiltonian system of the form x^′ = f(x, y), y^′ = g(x, y), where f(x, y) and g(x, y) are functions with parameters. We estimate the number of positive and oscillatory solutions for the boundary value problems. Our primary tool is the phase plane analysis combined with evaluations of time map functions. Multiple positive solutions are detected due to multiple period annuli.