The Stability of the Equilibrium of Planar Hamiltonian and Reversible Systems

In this paper, we study the stability of the equilibrium of planar systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^\prime = X(x,y,t), \quad y^\prime = Y(x,y,t),$$\end{document} where X and Y are real analytic in x, y, and t, and quasi-periodic in t with frequencies (ω1,...,ωn). Under some reasonable assumptions, we obtain a sufficient and necessary condition.