  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.
