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In this project we do : - start with the review of the theory of Ordinary Differential Equations (in short ODEs) and study an abstract Cauchy problem (in Banach spaces). - Study linear Systems, the notion of Stability and also consider nonlinear systems, - Introduce Hamiltonian systems in euclidean spaces and Hilbert spaces; - Show the existence of solutions for every finite dimensional autonomous Hamiltonian systems with initial conditions and Hamiltonian H of class C2 with bounded gradient. - Give a result by BENCI on the existence of periodic solutions of Hamiltonian systems (with prescribed energy h) with a regular Hamiltonian.