  We simulate the magnetization dynamics of a permalloy spheroid of nanoscopic
size in zero external field, such that both dipolar and exchange interactions
are important. Low excitation power is used to obtain the frequencies and mode
patterns of many normal modes. At higher power, non-linear three and four mode
couplings between magnons carrying orbital angular momentum are observed to
give rise to Suhl instabilities. Suhl's analysis of the selection rules
governing the allowed processes is extended to initial states other than
uniform precession. These rules are studied and confirmed by the simulations.
Both down- and up-conversion are seen as well as three and four-mode processes.
General trends are inferred for preferred instabilities among those that are
allowed, although the thresholds for some instabilities appear to be very high.
