On the reducibility of a class of almost periodic Hamiltonian systems

Jia Li; Junxiang Xu

doi:10.3934/dcdsb.2020268

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2021, Volume 26, Issue 7: 3905-3919. Doi: 10.3934/dcdsb.2020268

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On the reducibility of a class of almost periodic Hamiltonian systems

Jia Li^a, and
Junxiang Xu^b,

a.
School of Mathematics and Statistics, Xuzhou Institute of Technology, Xuzhou, 221111, China
b.
College of Mathematics, Southeast University, Nanjing, 210096, China

Received: October 2019

Revised: July 2020

Early access: September 2020

Published: July 2021

The authors are supported by NSFC grant 11526177 and the Natural Science Foundations for Colleges and Universities in Jiangsu Province grant 18KJB110029.

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Abstract

In this paper we consider the following linear almost periodic hamiltonian system

$ \dot{x} = (A+\varepsilon Q(t, \varepsilon))x, \; x\in R^{2}, $

where $ A $ is a constant matrix with different eigenvalues, and $ Q(t, \varepsilon) $ is analytic almost periodic with respect to $ t $ and analytic with respect to $ \varepsilon $. Without any non-degeneracy condition, we prove that the linear hamiltonian system is reducible for most of sufficiently small parameter $ \varepsilon $ by an almost periodic symplectic mapping.

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Mathematics Subject Classification: 37J40.

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