The stability of the elliptic equilibrium of planar quasi-periodic Hamiltonian systems

In this paper, we study the planar Hamiltonian system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot x = J(A(\theta )x + \nabla f(x,\theta )), \dot \theta = \omega , x \in \mathbb{R}^2 , \theta \in \mathbb{T}^d ,$$\end{document} where f is real analytic in x and θ, A(θ) is a 2 × 2 real analytic symmetric matrix, J = (1−1) and ω is a Diophantine vector. Under the assumption that the unperturbed system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot x = JA(\theta )x$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot \theta = \omega$$\end{document} is reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system.