# Periodic solutions of semilinear Duffing equations with impulsive effects

## TL;DR

This paper investigates the existence and multiplicity of periodic solutions in semilinear Duffing equations with impulsive effects, revealing differences between autonomous and nonautonomous cases using the Poincaré-Birkhoff twist theorem.

## Contribution

It extends the application of the Poincaré-Birkhoff twist theorem to impulsive Duffing equations, establishing the existence of infinitely many solutions in autonomous cases and finitely many in nonautonomous cases.

## Key findings

- Infinitely many periodic solutions for autonomous impulsive Duffing equations.
- Finitely many periodic solutions for nonautonomous impulsive Duffing equations.
- Impulses significantly affect the number of solutions compared to non-impulsive equations.

## Abstract

In this paper we are concerned with the existence of periodic solutions for semilinear Duffing equations with impulsive effects. Firstly for the autonomous one, basing on Poincar\'{e}-Birkhoff twist theorem, we prove the existence of infinitely many periodic solutions. Secondly, as for the nonautonomous case, the impulse brings us great challenges for the study, and there are only finitely many periodic solutions, which is quite different from the corresponding equation without impulses. Here, taking the autonomous one as an auxiliary equation, we find the relation between these two equations and then obtain the result also by Poincar\'{e}-Birkhoff twist theorem.