On a perturbation theory of Hamiltonian systems with periodic coefficients

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 ${\displaystyle \left(\widetilde{X}(t)\right)_{t\ge 0}}$ of the rank $k$ perturbation of Hamiltonian system with periodic coefficients and the rank $k$ perturbation of the fundamental matrix ${\displaystyle \left(X(t)\right)_{t\ge 0}}$ of the unperturbed system are the same, the Jordan canonical form of ${\displaystyle \left(\widetilde{X}(t)\right)_{t\ge 0}}$ 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.