**AUTHORS:**Mouhamadou Dosso, Arouna G. Y. Traore, Jean-Claude Koua Brou

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From a theory developed by C. Mehl, et al., a theory of the rank one perturbation of Hamiltonian systems with periodic coefficients is proposed. It is showed that the rank one perturbation of the fundamental solution of Hamiltonian system with periodic coefficients is solution of its rank one perturbation. Some results on the consequences of the strong stability of these types of systems on their rank one perturbation is proposed. Two numerical examples are given to illustrate this theory. 2010 Mathematics Subject Classification : 15A63, 15A21, 47A55, 93B10, 93C73.

**KEYWORDS:**
Eigenvalue, symplectic matrix, Hamiltonian system, Fundamental solutions, Perturbation

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