# American Institute of Mathematical Sciences

January  2019, 39(1): 431-445. doi: 10.3934/dcds.2019017

## An application of Moser's twist theorem to superlinear impulsive differential equations

 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author: Xiong Li

Received  March 2018 Revised  March 2018 Published  October 2018

Fund Project: The second author is supported by the NSFC (11571041) and the Fundamental Research Funds for the Central Universities

In this paper, we consider a simple superlinear Duffing equation
 $x''+2x^{3}+p(t) = 0\;\;\;\;\;\;\;\;(0.1)$
with impulses, where
 $p(t+1) = p(t)$
is an integrable function in
 $\mathbb{R}$
. In order to apply Moser's twist theorem, we need to ensure that the corresponding Poincaré map of (0.1) is quite close to a standard twist map but it is not usually achieved due to the existence of impulses. Two types of impulsive functions which overcome this problem with different effects in the Poincaré map are provided here. In both cases, there are large invariant curves diffeomorphism to circles surrounding the origin and going to the infinity, which confine the solutions in its interior and therefore lead to the boundedness of all solutions. Furthermore, it turns out that the solutions starting at
 $t = 0$
on the invariant curves are quasiperiodic.
Citation: Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017
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