Bounding Solutions of a Forced Oscillator

TL;DR

This paper presents a mathematical proof establishing the boundedness of solutions for a forced oscillator using advanced techniques like action-angle variables, Lie transformations, and Moser's invariant curve theorem.

Contribution

It generalizes Morris' original theorem by applying a convergent Lie transformation and invariant curve theorem to prove boundedness in a broader setting.

Findings

Proof of boundedness for a forced oscillator.

Extension of Morris' theorem to a more general case.

Use of Lie transforms to reduce complex cases to simpler ones.

Abstract

Our proof is based on a generalization of action-angle variables, a convergent Lie transformation, and Moser's invariant curve theorem. As an overall outline we give a quick proof of Morris' original theorem. Then the full theorem is established in two steps. In the first step we define a special case and then follow the proof given for Morris' Theorem to prove boundedness for the special case. In the second step we use the method of Lie transforms to reduce the general case to the special case.