Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance (Q1948775)

The paper deals with a planar Hamiltonian system of the form \[ Jz'=\nabla H(z)+f(z)+p(t), \] where \(f: \mathbb R^2 \to \mathbb R^2\) is locally Lipschitz continuous and \(p:\mathbb R \to \mathbb R^2\) is continuous and \(2\pi\)-periodic, the Hamiltonian \(H\) has a locally Lipschitz gradient and is positively homogeneous of degree 2 and positive. A result on the coexistence of unbounded solutions and periodic solutions is obtained under weaker assumptions than those in [\textit{A. Fonda} and \textit{J. Mawhin}, Adv. Differ. Equ. 11, No. 10, 1111--1133 (2006; Zbl 1155.34020)]. The main tool is a careful analysis of the dynamics of the Poincaré map of the system.