Lagrange stability for asymmetric Duffing equations (Q1590097)

For the Duffing equation \(\frac{d^2x}{d^2t}+g(x)=e(t)\) where \(e(t)\) is of period 1 and \(g(x)\text{sgn }x\rightarrow +\infty \) as \(x\rightarrow \pm \infty\) let us associate \(a=\lim_{x \to -\infty } \frac{g(x)}{x}\) and \(b=\lim_{x \to +\infty } \frac{g(x)}{x}\). Using Moser's twist theorem, the Lagrange stability in the cases \(-\infty <a<+\infty\), \(b=+\infty\) and \(-\infty <b<+\infty\), \(a=-\infty\) is studied.