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

TL;DR

This paper establishes a new sufficient condition for the existence of periodic solutions in superquadratic Hamiltonian systems, even allowing finite-time blow-up, by leveraging rotational properties without growth restrictions.

Contribution

It introduces a novel fixed-point method that does not require flow growth conditions, broadening the class of systems where periodic solutions can be guaranteed.

Findings

Proves existence of periodic solutions under new conditions.

Applies to systems with finite-time blow-up.

Provides examples illustrating finite-time blow-up phenomena.

Abstract

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.