  Non-linear magnetization dynamics is essential for the operation of many
spintronics devices. For microwave assisted switching of magnetic elements the
low field regime is of particular interest. In addition a large number of
experiments uses high amplitude FMR in order to generate d.c. currents via spin
pumping mechanism. Here we use time resolved X-ray magnetic circular dichroism
experiments to determine the number density of excited magnons in magnetically
soft Ni_80Fe_20 thin films at small bias fields and large rf-excitation
amplitudes. Our data shows that the common model of non-linear ferromagnetic
resonance is not suitable to describe the low bias field limit. Here we derive
a new model of parametric spin-wave excitation which correctly predicts
threshold amplitudes and decay rates also at low bias fields. In fact a new
series of critical modes with amplitude phase oscillations is found,
generalizing the theory of parametric spin-wave excitation.
