On the reducibility of a class of almost periodic Hamiltonian systems
type: Item
instance of: scholarly article
title: On the reducibility of a class of almost periodic Hamiltonian systems
zbMATH Open document ID: 1477.37067
author: J. Li
author: Junxiang Xu
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2021-06-17T00:00:00Z
review text: A linear system of the form \[ \dot{x} = A(t)x ,\qquad x\in{\mathbb{R}^{n}}, \] is reducible if there exists a quasi-periodic non-singular transformation \(x = \phi(t)y\), where \(\phi(t)\) and \(\phi(t)^{-1}\) are quasi-periodic and bounded, such that \(\dot{x} = A(t)x\) is transformed into \[\dot{y} = By,\qquad y\in{\mathbb{R}^{n}}, \] where \(B\) is a constant matrix. Here the authors consider the system \[ \dot{x} = (A + \epsilon Q(t,\epsilon))x , \qquad x\in{\mathbb{R}^{2}},\] where \(A\) is a constant matrix with distinct eigenvalues, and \(Q(t,\epsilon)\) is analytic and almost periodic with respect to \(t\) and analytic with respect to \(\epsilon\). Without any non-degeneracy condition, it is proved that the system is reducible for any choice of a sufficiently small parameter \(\epsilon\) by an almost periodic symplectic mapping.
reviewed by: Tudor Bînzar
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37J06
Mathematics Subject Classification ID: 37J46
Mathematics Subject Classification ID: 37J11
zbMATH DE Number: 7360608
zbMATH Keywords: almost periodic
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: non-degeneracy condition
zbMATH Keywords: KAM theory
MaRDI profile type: Publication
full work available at URL: dcdsb.2020268
OpenAlex ID: W3084173331
cites work: Q4097594
cites work: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation
cites work: Full measure reducibility for generic one-parameter family of quasi-periodic linear systems
cites work: Smoothness of spectral subbundles and reducibility of quasi periodic linear differential systems
cites work: On the reducibility of linear differential equations with quasiperiodic coefficients
cites work: On Quasi-Periodic Perturbations of Elliptic Equilibrium Points
cites work: On the reducibility of a class of finitely differentiable quasi-periodic linear systems
cites work: On the reducibility of a class of quasi-periodic Hamiltonian systems with small perturbation parameter near the equilibrium
cites work: Small divisors with spatial structure in infinite dimensional Hamiltonian systems
cites work: Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities
cites work: On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point
cites work: Q4237775
cites work: On the reducibility of a class of linear differential equations with quasiperiodic coefficients
cites work: On the reducibility of two-dimensional linear quasi-periodic systems with small parameter
cites work: On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters
cites work: Persistence of invariant torus in Hamiltonian systems with two-degree of freedom
DOI: 10.3934/DCDSB.2020268
description: scientific article; zbMATH DE number 7360608