Planar Hamiltonian systems: index theory and applications to the existence of subharmonics

We consider a planar Hamiltonian system of the type $Jz' = \nabla_z H(t,z)$, where $H: \mathbb{R} \times \mathbb{R}^2 \to \mathbb{R}$ is a function periodic in the time variable, such that $\nabla_z H(t,0) \equiv 0$ and $\nabla_z H(t,z)$ is asymptotically linear for $\vert z \vert \to +\infty$. After revisiting the index theory for linear planar Hamiltonian systems, by using the Poincar\'e-Birkhoff fixed point theorem we prove that the above nonlinear system has subharmonic solutions of any order $k$ large enough, whenever the rotation numbers (or, equivalently, the mean Conley-Zehnder indices) of the linearizations of the system at zero and at infinity are different. Applications are given to the case of planar Hamiltonian systems coming from second order scalar ODEs.