  The largest part of any gravitational-wave inspiral of a compact binary can
be understood as a slow, adiabatic drift between the trajectories of a certain
referential conservative system. In many contexts, the phase space of this
conservative system is smooth and there are no "topological transitions" in the
phase space, meaning that there are no sudden qualitative changes in the
character of the orbital motion during the inspiral. However, in this chapter
we discuss the cases where this assumption fails and non-linear and/or
non-smooth transitions come into play. In integrable conservative systems under
perturbation, topological transitions suddenly appear at resonances, and we
sketch how to implement the passage through such regions in an inspiral model.
Even though many of the developments of this chapter apply to general
inspirals, we focus on a particular scenario known as the Extreme mass ratio
inspiral (EMRI). An EMRI consists of a compact stellar-mass object inspiralling
into a supermassive black hole. At leading order, the referential conservative
system is simply geodesic motion in the field of the supermassive black hole
and the rate of the drift is given by radiation reaction. In Einstein gravity
the supermassive black hole field is the Kerr space-time in which the geodesic
motion is integrable. However, the equations of motion can be perturbed in
various ways so that prolonged resonances and chaos appear in phase space as
well as the inspiral, which we demonstrate in simple physically motivated
examples.
