  Controlling non-linear effects in the transverse dynamics of charged particle
beams in circular accelerators opens new possibilities for controlling some of
the beam properties. Beam splitting by crossing a stable 1D non-linear
resonance is part of the routine operation of the CERN Proton Synchrotron. The
beam undergoes trapping and transport inside stable islands created in the
horizontal plane to allow multi-turn extraction towards the Super Proton
Synchrotron, where the beam is used for fixed-target experiments. This process
acts only on the horizontal beam emittance, inducing a reduction of its initial
value. In this paper, we present a generalisation of this approach, in which
both transverse planes are affected by the proposed technique. We will discuss
in detail how to manipulate the transverse emittances by means of a controlled
crossing of a 2D non-linear resonance. The novel technique will be presented by
discussing the theoretical analysis of a Hamiltonian model, as well as
simulating the performance of the proposed manipulation using a more realistic
non-linear symplectic map.
