Unusual fourth order deformation of beam due to space charge

Unusual fourth order core deformation in y-phase space is observed in the simulation of SNS ring without lattice imperfections. The fourth order core deformation happens for a wide range of tunes, whereas an asymmetric deformation takes place when bare tune v_y is 5.53 to 5.74. When v_y is close to 6, 2:1 resonance appears in both planes. Extensive numerical simulation indicates that the difference resonance 2_x-2v_y = 0 from space charge potential and dispersion effect are playing a crucial role. Similar effect is expected near the resonance 2v_x 2v_y=P with a superperiod P. This is unusual because the SNS ring under model has a perfect four-fold symmetry and operating points are far from structural resonances. Also an interesting observation on equipartitioning is presented in section 4