Boundedness of solutions in a class of Duffing equations with a bounded restore force
The equation \[ \frac{d^2x}{dt^2}+\arctan(x)=p(t), \] with \(p\in C^5\), \(p(t)=p(t+1)\), \(|\int^1_0p(t)\,dt|<\frac{\pi}{2}\), is considered. The boundedness of solutions, i.e., \(\sup_{t\in \mathbb{R}}(|x(t)|,|x'(t)|)<+\infty\), is proved by a series of transformations leading the equation to an appropriate form of KAM theory.
scientific article; zbMATH DE number 5015926