Lagrange stability for impulsive Duffing equations

TL;DR

This paper proves the boundedness of solutions for impulsive Duffing equations with polynomial potentials using KAM theory, showing the existence of many quasi-periodic solutions and extending classical results to impulsive cases.

Contribution

It extends classical boundedness results of Duffing equations to impulsive cases with low regularity, using KAM theorem to establish solution boundedness and quasi-periodic solutions.

Findings

Solutions are bounded for all time under certain impulses.

Many quasi-periodic solutions exist with positive measure.

Results extend classical Duffing equation properties to impulsive systems.

Abstract

This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity. This result extends some well-known results on Duffing equations to impulsive Duffing equations.