Resonant instability from synchrobetatron coupling or tune modulation

Different mechanisms may generate nonlinear resonances that can be seen as sidebands of the pure betatron resonances. Two of them, proceeding from the same principle, are dealt with in this paper. The first one concerns the coupling of the equations of betatron motion and synchrotron motion through the chromatic dependence of the transverse tune with energy. The second one arises from tune modulation due for instance to power supply ripple. Both result in a time variation of the phase advance and the treatment proposed applies to them identically. It gives a criterion to distinguish between stability and amplitude growth or instability. Application to design studies of accelerators is given.