  We develop a theory for the magnon Kerr effect in a cavity magnonics system,
consisting of magnons in a small yttrium iron garnet (YIG) sphere strongly
coupled to cavity photons, and use it to study the bistability in this hybrid
system. To have a complete picture of the bistability phenomenon, we analyze
two different cases in driving the cavity magnonics system, i.e., directly
pumping the YIG sphere and the cavity, respectively. In both cases, the magnon
frequency shifts due to the Kerr effect exhibit a similar bistable behavior but
the corresponding critical powers are different. Moreover, we show how the
bistability of the system can be demonstrated using the transmission spectrum
of the cavity. Our results are valid in a wide parameter regime and generalize
the theory of bistability in a cavity magnonics system.
