  The beam-plasma instability can be addressed as a reduced model in several
contexts of plasma physics, from space to fusion plasma. In this paper, we
review and refine some non-linear features of this model. Specifically, by
analyzing the dependence of the non-linear velocity spread as a function of the
linear growth rate, we discuss the effective size of the resonance in view of
its role in the spectral overlap at saturation. The relevance of this
characterization relies on the necessity of a quantitative determination of the
overlap degree to discriminate among different transport regimes of the self
consistent dynamics. The analysis is enriched with a study of the phase-space
dynamics by means of the Lagrangian Coherent Structure technique, in order to
define the transport barriers of the system describing the relevant features of
the overlap process. Finally, we discuss relevant features related to the mode
saturation levels.
