Dynamics of Slow-Fast Hamiltonian Systems: The Saddle–Focus Case

We study the dynamics of a multidimensional slow-fast Hamiltonian system in a neighborhood of the slow manifold under the assumption that the frozen system has a hyperbolic equilibrium with complex simple leading eigenvalues and there exists a transverse homoclinic orbit. We obtain formulas for the corresponding Shilnikov separatrix map and prove the existence of trajectories in a neighborhood of the homoclinic set with a prescribed evolution of the slow variables. An application to the $$3$$ body problem is given.