On a perturbation theory of Hamiltonian systems with periodic coefficients

Abstract

A theory of rank $k\ge 2$ perturbation of symplectic matrices and Hamiltonian systems with periodic coefficients using a base of isotropic subspaces, is presented. After showing that the fundamental matrix ${\left(\tilde{X}(t)\right)}_{t\ge 0}$ of the rank $k$ perturbation of Hamiltonian system with periodic coefficients and the rank $k$ perturbation of the fundamental matrix ${\left(X(t)\right)}_{t\ge 0}$ of the unperturbed system are the same, the Jordan canonical form of ${\left(\tilde{X}(t)\right)}_{t\ge 0}$ is given. Two numerical examples illustrating this theory and the consequences of rank $k$ perturbations on the strong stability of Hamiltonian systems were also given.