Existence of a periodic solution for superquadratic Hamiltonian systems with possible finite-time blow-up

We prove a sufficient condition for the existence of a $T$-periodic solution for the planar system $\dot{z}=F(t,z)$, characterized by the growth to infinity of the rotations made in one period by solutions starting at increasingly large initial values. Our result applies in particular to superquadratic Hamiltonian systems satisfying the Ambrosetti–Rabinowitz condition. The key novelty of the paper is that we do not require any growth condition on the flow to ensure global existence of solutions, allowing finite-time blow-up. Our method is based on a fixed-point theorem which exploits the rotational properties of the dynamics. To conclude, we discuss a family of examples of Hamiltonian systems showing finite-time blow-up.