Invariant Tori of Impulsive Duffing-Type Equations via KAM Technique

TL;DR

This paper applies KAM techniques to impulsive Duffing-type equations to establish the existence of invariant tori and quasiperiodic solutions, demonstrating boundedness and stability under impulsive perturbations.

Contribution

It introduces a novel KAM-based method for impulsive equations, proving the existence of invariant tori and quasiperiodic solutions with area-preserving impulsive terms.

Findings

Existence of invariant circle for impulsive Duffing equations.

Solutions on the invariant circle are quasiperiodic.

Boundedness and Lagrange stability of solutions.

Abstract

A method via the KAM technique is introduced to study the existence of invariant tori and quasiperiodic solutions for impulsive Duffing-type equations with time period 1. Basing on several planar symplectic homeomorphisms and some estimates of impulsive perturbations under each symplectic homeomorphisms, we prove via the Moser's twist theorem the boundedness (Lagrange stability) and the existence of an invariant circle for the equation with area-preserving impulsive terms. And this invariant circle having any rotation number $\omega>\omega_0$ with some $\omega_0>0$, so we obtain also that the solutions starting from the circle are quasiperiodic with frequencies $\omega$ and 1.