Abstract. 
I will describe two different settings where geometrical methods can be
applied to detect (global) instability in mechanical systems: a priori
chaotic and a priori unstable Hamiltonian systems. A very wide class of
geodesic flows in any dimension plus a quasi-periodic perturbation give
rise to a priori chaotic Hamiltonian system, whereas a priori unstable
Hamiltonian systems take place in considering periodic perturbations of a
(or some) pendulum plus a (or some) rotor.

In both cases, there is a very big invariant object called NHIM (normally
hyperbolic invariant manifold), which apart from its inner dynamics,
possesses an outer dynamics, due to the transversal intersection of its
associated unstable and stable invariant manifolds, which is described by
the so called scattering map.

The combination of both dynamics along the NHIM gives rise to chaotic and
unstable global behavior.

This talk is based on joint work with Gemma Huguet, Rafael de la
Llave and Tere M. Seara.