Recently, a novel multi-turn extraction scheme was proposed, based on particle trapping inside stable resonances. Numerical simulations and experimental tests confirmed the feasibility of such a scheme for low order resonances. While the 3rd order resonance is generically unstable and those higher than 4th order are generically stable, the 4th order resonance can be either stable or unstable depending on the details of the system under consideration. By means of the normal form approach a general formula to control the stability of the 4th order resonance is derived. Numerical simulations confirm the analytical results and show that by crossing the unstable 4th order resonance the region around the centre of phase space is depleted and particles are trapped only in the four stable islands. This indicates that a four-turn extraction could be envisaged based on this technique.
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