The Poincaré–Birkhoff Theorem for a Class of Degenerate Planar Hamiltonian Systems

Abstract

In this paper, we investigate the problem of the existence and multiplicity of periodic solutions to the planar Hamiltonian system x' = −λα(t)f (y), y' = λβ(t)g(x), where α, β are non-negative T-periodic coefficients and λ > 0. We focus our study to the so-called “degenerate” situation, namely when the set Z := supp α ∩ supp β has Lebesgue measure zero. It is known that, in this case, for some choices of α and β, no nontrivial T-periodic solution exists. On the opposite, we show that, depending of some geometric configurations of α and β, the existence of a large number of T-periodic solutions (aswell as subharmonic solutions) is guaranteed (for λ > 0 and large). Our proof is based on the Poincaré–Birkhoff twist theorem. Applications are given to Volterra’s predator-prey model with seasonal effects.