  In a recent paper, the adiabatic theory of Hamiltonian systems was
successfully applied to study the crossing of the linear coupling resonance,
$Q_x-Q_y=0$. A detailed explanation of the well-known phenomena that occur
during the resonance-crossing process, such as emittance exchange and its
dependence on the adiabaticity of the process was obtained. In this paper, we
consider the crossing of the resonance of nonlinear coupling $2 Q_x -2 Q_y = 0$
using the same theoretical framework. We perform the analysis using a
Hamiltonian model in which the nonlinear coupling resonance is excited and the
corresponding dynamics is studied in detail, in particular looking at the
phase-space topology and its evolution, in view of characterizing the emittance
exchange phenomena. The theoretical results are then tested using a symplectic
map. Thanks to this approach, scaling laws of general interest for applications
are derived.
