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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 periodicin the time variable, such that $\nabla_z H(t,0) \equiv 0$ and $\nabla_zH(t,z)$ is asymptotically linear for $\vert z \vert \to +\infty$. Afterrevisiting the index theory for linear planar Hamiltonian systems, by using thePoincar\'e-Birkhoff fixed point theorem we prove that the above nonlinearsystem has subharmonic solutions of any order $k$ large enough, whenever therotation numbers (or, equivalently, the mean Conley-Zehnder indices) of thelinearizations of the system at zero and at infinity are different.Applications are given to the case of planar Hamiltonian systems coming fromsecond order scalar ODEs.

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