Nonlinear Stability of Equilibria in Hamiltonian Systems with Multiple Resonances without Interactions

Abstract

In this work, we advance in the study of the Lyapunov stability and instability of equilibrium solutions of Hamiltonian flows. More precisely, we study the nonlinear stability in the Lyapunov sense of equilibrium solutions in autonomous Hamiltonian systems with n-degrees of freedom, assuming the existence of two resonance vectors k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{k}^1$$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{k}^2$$\\end{document} without interaction (|k1|≤|k2|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\ extbf{k}^1|\\le |\ extbf{k}^2|$$\\end{document}). We provide conditions to obtain a type of formal stability, called Lie stability. In particular, we need to normalize the Hamiltonian function to any arbitrary order, and our results take into account the sign of the components of the resonance vectors. Subsequently, we guarantee some sufficient conditions to obtain exponential stability in the sense of Nekhoroshev for Lie stable systems. In addition, we give sufficient conditions for the instability in the Lyapunov sense of the full system. For this, it is necessary to normalize the Hamiltonian function to an adequate order, and assuming that the components of at least one resonance vector change of sign.