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

TL;DR

This paper develops an index theory approach for planar Hamiltonian systems with periodic Hamiltonians, proving the existence of subharmonic solutions of any large order when certain rotation number conditions are met, with applications to scalar ODEs.

Contribution

It introduces a novel index theory framework for analyzing subharmonic solutions in planar Hamiltonian systems with asymptotic linearity, extending previous results using the Poincaré-Birkhoff theorem.

Findings

Existence of subharmonic solutions of arbitrary large order under specific rotation number conditions.

Application of index theory to systems derived from second order scalar ODEs.

Demonstration of the method's effectiveness through concrete examples.

Abstract

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.