Invariant tori in nonlinear oscillations
The boundedness of all solutions to the semilinear Duffing equation \[ x''+ \omega^2x+ \phi(x)= p(t),\quad \omega\in \mathbb{R}^+\setminus \mathbb{N}, \] where \(P(t)\) is a smooth \(2\pi\)-periodic function and \(\phi(x)\) is bounded, is investigated. Typically, the authors show that the solutions are all bounded if, additionally, \(\lim_{x\to\pm \infty} \phi(x)\) are both finite and different, and \(\lim_{x\to\pm\infty}x^5 \phi^{(5)}(x)= 0\). The proof of this theorem requires eight technical lemmas.
scientific article; zbMATH DE number 1439355