High power microwave experiments conducted on a rectangular yttrium-iron-garnet (YIG) film indicate a parabolic dependence of the square of the frequency of auto-oscillations on microwave signal amplitudes, an observation that is qualitatively consistent with prior theoretical predictions. Forward volume magnetostatic Waves were excited using a microstrip transducer kept in contact with a YIG film placed in a constant external magnetic field (H-dc = 3.48 kG). Variations in the input microwave power (10 dBm < P < 25 dBm) and frequency (5.1 GHz < f < 5.7 GHz) were used to locate and study auto-oscillations close to the Suhl instability at the main resonance. A novel method of viewing changes in the microwave passband using density plots enables us to study variations in the dipole gaps in the passband near the Suhl instability. A broadening of dipole gaps into fingerlike regions of weak transmission marks the onset of auto-oscillations beyond the instability threshold. These regions are associated with a variety of spin-wave dynamics ranging from simple auto-oscillations to auto-oscillations with period doubling and in some cases an abrupt transition to a turbulent wide-band power spectrum. The parabolic dependence of the auto-oscillation frequency persists despite a period-doubling bifurcation. (C) 1996 American Institute of Physics.