  We show that a self-interacting neutrino gas can spontaneously acquire a
non-stationary pulsating component in its flavor content, with a frequency that
can exactly cancel the "multi-angle" refractive effects of dense matter. This
can then enable homogeneous and inhomogeneous flavor conversion instabilities
to exist even at large neutrino and matter densities, where the system would
have been stable if the evolution were strictly stationary. Large flavor
conversions, especially close to a supernova core, are possible via this novel
mechanism. This may have important consequences for the explosion dynamics,
nucleosynthesis, as well as for neutrino observations of supernovae.
