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Homoclinic orbits near saddle-center fixed points of Hamiltonian systems with two degrees of freedom

We study a class of Hamiltonian systems on a 4 dimensional symplectic manifold which have a saddle-center fixed point and satisfy the following property: All the periodic orbits in the center manifold of the fixed point have an orbit homoclinic to them, although the fixed point itself does not. In addition, we prove that these systems have a chaotic behavior in the neighborhood of the energy shell of the fixed point.