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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document} body problem is given.