  We theoretically explore nonlinearities of ferromagnets in microwave cavities
in the classical and quantum regimes, and assess the resources for quantum
information, i.e. fluctuation squeezing and bipartite entanglement. The
(semi-)classical analysis of the anharmonic oscillator (Duffing) model for the
Kittel mode when including all other magnon modes, reveals chaotic and
limit-cycle phases that do not survive in quantum calculations. However,
magnons with nonzero wavenumbers that are driven by the Suhl instability of the
Kittel mode, form a genuine limit cycle. We subsequently compute bounds for the
distillable entanglement, as well as entanglement of formation for the
bipartite configurations of the mixed magnon modes. The distillable
entanglement of bipartite states accessible from a covariance matrix vanishes,
but can be recovered by injection locking. The predicted magnon entanglement
can be experimentally tested with yttrium iron garnet samples under realistic
conditions.
