| Abstract: |
| \noindent Periodic solutions for superlinear planar systems have been extensively studied, in particular in the Hamiltonian setting, for instance in the work of A. Boscaggin (2012). Such results typically rely on the global existence of solutions, a structural assumption whose role was clarified by P. Gidoni (2023) for a class of superlinear second-order equations where global existence may fail. In this paper we extend the results of Gidoni (2023) to general non-autonomous $T$-periodic planar systems of the form
\begin{equation*}
\dot{z} = F(t,z),
\end{equation*}
without assuming global forward existence of solutions. Under suitable asymptotic bounds on the rotation number of solutions, we prove the existence of at least one $T$-periodic solution.
In the Hamiltonian case $F(t,z)=J\nabla_z H(t,z)$, we further investigate super-quadratic systems and provide explicit sufficient conditions ensuring that the rotational assumptions are satisfied.
This is a joint work with Paolo Gidoni (University of Udine). |
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