Analysis of positive measure reducibility for quasi-periodic linear systems under Brjuno-Rüssmann condition

Muhammad Afzal; Tariq Ismaeel; Riaz Ahmad; Ilyas Khan; Dumitru Baleanu; Muhammad Afzal; Tariq Ismaeel; Riaz Ahmad; Ilyas Khan; Dumitru Baleanu

doi:10.3934/math.2022520

AIMS Mathematics

2022, Volume 7, Issue 5: 9373-9388. doi: 10.3934/math.2022520

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Analysis of positive measure reducibility for quasi-periodic linear systems under Brjuno-Rüssmann condition

1.
Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan
2.
Department of Mathematics, Government College University, Lahore 54000, Pakistan
3.
Faculty of Science, Yibin University, Yibin 644000, China
4.
Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah, P.O. Box 66, Majmaah 11952, Saudi Arabia
5.
Department of Mathematics, Cankaya University, Ankara 06790, Turkey
6.
Institute of Space Sciences, 077125 Magurele, Romania
7.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan

Received: 30 November 2021 Revised: 07 February 2022 Accepted: 16 February 2022 Published: 11 March 2022
MSC : 37K55, 70K40

In this article, we discuss the positive measure reducibility for quasi-periodic linear systems close to a constant which is defined as:

$ \begin{align*} \frac{dx}{dt} = (A(\lambda) + Q(\varphi,\lambda))x, \dot{\varphi} = \omega, \end{align*} $

where $ \omega $ is a Brjuno vector and parameter $ \lambda\in (a, b) $. The result is proved by using the KAM method, Brjuno-Rüssmann condition, and non-degeneracy condition.
- quasi-periodic,
- Brjuno-Rüssmann condition,
- reducibility,
- KAM method
Citation: Muhammad Afzal, Tariq Ismaeel, Riaz Ahmad, Ilyas Khan, Dumitru Baleanu. Analysis of positive measure reducibility for quasi-periodic linear systems under Brjuno-Rüssmann condition[J]. AIMS Mathematics, 2022, 7(5): 9373-9388. doi: 10.3934/math.2022520

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Abstract

In this article, we discuss the positive measure reducibility for quasi-periodic linear systems close to a constant which is defined as:

$ \begin{align*} \frac{dx}{dt} = (A(\lambda) + Q(\varphi,\lambda))x, \dot{\varphi} = \omega, \end{align*} $

where $ \omega $ is a Brjuno vector and parameter $ \lambda\in (a, b) $. The result is proved by using the KAM method, Brjuno-Rüssmann condition, and non-degeneracy condition.

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