Virtual cathode formation in electron beams as a bifurcation phenomena

The virtual cathode represents a stable oscillatory state for electron beams which exceed the space-charge limit. It originates at the convergence point of two unstable steady states with the stable oscillatory state. The scaling parameters for oscillation frequency, potential minimum and position are based on beam current and momentum. Two species, electron and ion, space-charge flow stability is also explained in terms of bifurcation theory. The Pierce instability is a special case of this broader class.