Stability of periodic Hamiltonian systems with equal dissipation

This contribution highlights that a linear periodic Hamiltonian system preserves a symplectic structure if a particular dissipation is present. This specific structure is defined by the algebraic properties of mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-symplectic matrices and symmetry of its eigenvalues. A method is established for the stability analysis of this class of systems consisting of damped and coupled Mathieu equations. It enables an efficient computation of the corresponding stability chart. One main strength of the method is the calculation of the stability chart even for large parameter values, especially for the amplitude of the parametric excitation and the system response itself. The proposed stability analysis is applied in detail on two examples consisting of two coupled equations.