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Non-perturbative reducibility of three-dimensional skew symmetric systems without any non-degeneracy condition

Part of: Hamiltonian and Lagrangian mechanics Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  18 February 2025

DONGFENG ZHANG Open the ORCID record for DONGFENG ZHANG [Opens in a new window]  and
JUNXIANG XU Open the ORCID record for JUNXIANG XU [Opens in a new window]
DONGFENG ZHANG*
Affiliation:
School of Mathematics, Southeast University, Nanjing 210096, PR China
JUNXIANG XU
Affiliation:
School of Mathematics, Southeast University, Nanjing 210096, PR China
*
e-mail: zhdf@seu.edu.cn
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Abstract

We consider the system $\dot {x}=(A+ P(t,\epsilon )) x, x\in \mathbb {R}^{3}, $ where $A, P$ are both three-dimensional skew symmetric matrices, A is a constant matrix with eigenvalues $\pm i\unicode{x3bb} $ and 0, $P(t,\epsilon )$ is $C^{m}(m=0,1)$-smooth in $\epsilon $ and analytic quasi-periodic with respect to t with basic frequencies $\omega =(1,\alpha )$, with $\alpha $ being irrational, and $\epsilon $ is a small parameter. Under some non-resonant conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy condition, it is proved that for many sufficiently small parameters, this system can be reduced to a rotation system. Furthermore, if the basic frequencies satisfy that $ 0\leq \beta (\alpha ) < r,$ where $\beta (\alpha )=\limsup \nolimits _{n\rightarrow \infty } {\ln q_{n+1}}/{q_{n}},$ $q_{n}$ is the sequence of denominations of the best rational approximations for $\alpha \in \mathbb {R} \setminus \mathbb {Q},$ r is the initial radius of analytic domain, it is proved that for many sufficiently small parameters, this system can be reduced to a constant system.

Keywords

skew symmetric systemsreducibilityKAM iterationLiouvillean frequenciesnon-degeneracy condition

MSC classification

Primary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Secondary: 70H08: Nearly integrable Hamiltonian systems, KAM theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 37
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© The Author(s), 2025. Published by Cambridge University Press

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