An experiment involving the natural oscillations on an electron beam propagating parallel to a magnetic field in a low pressure gas is described. For conditions such that omega _pe²<< omega _ge² and beam currents above a few mu A, the beam system behaves as if it were approximately neutralized by cold ions. Increasing the beam current, I_b, increases the frequency and amplitude of the oscillation until period doubling occurs. With a further small increase, many more period doublings occur in rapid succession until the spectrum of the oscillations is essentially flat out to the original frequency f₀ above which it rapidly decreases. This state remains as I_b is increased until, suddenly, a perfectly coherent three cycle oscillation appears with a periodicity of three times that of f₀; with a small further increase of I_b, the three cycle becomes a 6 cycle and then the chaotic regime returns. The frequency f₀ agrees well with the ion plasma frequency of the neutralized electron beam and is probably driven by some form of shear instability. By considering only the amplitude of the oscillations the evolution of the system has been successfully modelled with a simple one dimensional difference equation which takes account of the competition between exponential growth of the instability and quadratic saturation leading to the final steady state.