Boundedness and unboundedness of solutions for reversible oscillators at resonance (Q5890380)

The paper concerns the ``non-Hamiltonian'' equation \(x''+f(x)x'+n^2x+\varphi(x)=p\) where \(p\) is \(2\pi\)-periodic, \(\lim_{x\to+\infty}\varphi(x)=\varphi(+\infty)\in \mathbb{R}\), etc. The main results state that, in the \((x,x')\)-phase plane, all solutions to the equation are bounded if \(4|\varphi(+\infty)|>|\int_0^{2\pi}p(t)\sin t dt|\) holds, some solutions are unbounded if the reverse inequality holds, and all solutions are unbounded under a sharper reverse inequality. The proofs use the fact that the associated Poincaré map satisfies the assumptions of a recent twist theorem for reversible systems [\textit{B. Liu}, Invariant curves of reversible mappings with small twist (preprint)].