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

We study the dynamics of a multidimensional slow-fast Hamiltonian system in a neighborhoodof the slow manifold under the assumption that the frozen system has a hyperbolic equilibriumwith complex simple leading eigenvaluesand there exists a transverse homoclinic orbit.We obtain formulas for the corresponding Shilnikov separatrix mapand prove the existence of trajectories in a neighborhood of the homoclinic setwith a prescribed evolution of the slow variables.An application to the \(3\) body problem is given.