On the Reducibility of a Class of Linear Almost Periodic Hamiltonian Systems

In this paper, we study the reducibility problem for a class of analytic almost periodic linear Hamiltonian systems dxdt=J[A+εQ(t)]x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{dx}{dt} = J[{A}+\varepsilon {Q}(t)]x \end{aligned}$$\end{document}where A is a symmetric matrix, J is an anti-symmetric symplectic matrix, Q(t) is an analytic almost periodic symmetric matrix with respect to t, and ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is a sufficiently small parameter. It is also assumed that JA has possible multiple eigenvalues and the basic frequencies of Q satisfy the non-resonance conditions. It is shown that, under some non-resonant conditions, some non-degeneracy conditions and for most sufficiently small ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , the Hamiltonian system can be reduced to a constant coefficients Hamiltonian system by means of an almost periodic symplectic change of variables with the same basic frequencies as Q(t).