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For a given a normally hyperbolic invariant manifold, whose stable andunstable manifolds intersect transversally, we consider several tools andtechniques to detect trajectories with prescribed itineraries: the scatteringmap, the transition map, the method of correctly aligned windows, and theshadowing lemma. We provide an user's guide on how to apply these tools andtechniques to detect unstable orbits in Hamiltonian systems. This consists inthe following steps: (i) computation of the scattering map and of thetransition map for a flow, (ii) reduction to the scattering map and to thetransition map, respectively, for the return map to some surface of section,(iii) construction of sequences of windows within the surface of section, withthe successive pairs of windows correctly aligned, alternately, under thetransition map, and under some power of the inner map, (iv) detection oftrajectories which follow closely those windows. We illustrate this strategywith two models: the large gap problem for nearly integrable Hamiltoniansystems, and the the spatial circular restricted three-body problem.

Transition map and shadowing lemma for normally hyperbolic invariant manifolds