Stability thresholds for transverse dipole modes with nonlinear space charge, chromaticity and octupoles

Transverse stability due to combinations of chromaticity effect, nonlinear space charge and octupoles of different polarities plays an important role in the determination of the impedance budget for the FAIR synchrotrons. Different analytic approaches have been suggested, for which no direct comparison has been made so far. In order to clarify this issue we perform numerical investigations employing the particle tracking code PATRIC and compare results of simulation scans with predictions of a dispersion relation.