Boundedness for semilinear Duffing equations at resonance (Q1758320)

The authors study the boundedness for semilinear Duffing equations at resonance. They deal with the equation \[ x''+n^2x+\phi(x)+g''(x)q(t)=0,\quad n\in \mathrm{N}. \] If the limits \(\lim_{x\to\pm}\phi(x)=\phi(\pm\infty)\) exist and are finite and \(\phi(+\infty))\neq\phi(-\infty)\) and if additionally, \(\lim_{|x|\to\infty}x^{12}\phi^{(12)}(x)=0\) and \(|g^{(k)}(x)|\leq C\), \(0\leq k\leq 14\), \(q\in C^{13}(\mathbb{R}/2\pi \mathbb{Z})\), they prove that all solutions of the given equation are bounded by using Moser's small twist theorem. The main idea is to take a series of canonical transformations under which the original system is transformed into a perturbation of an integrable Hamiltonian system outside of a large disc. The Poincaré map of the transformed system is close to a twist map. Then Moser's twist theorem guarantees the existence of arbitrarily large invariant curves diffeomorphic to circles. Every such curve confines the solutions in the interior and leads to a bound of these solutions.