Abstract: A 1D time-varying nonlinear theory based on the Duffing equation is applied to space-charge limited beams and specifically vircators. This theory classifies test-particle trajectories in a modulated nonlinear potential. Two predictions of the theory that can be directly compared to experiment are the final state of electron trajectories and the oscillation frequency of the electrons in the potential well. Experimental measurements of electron flux recorded along the vircator chamber wall correlates well with the numerically integrated final state of electron trajectory in the 1D theory. The oscillation frequency measured in the experiment is shown to be a better match to the oscillation frequency calculated from the nonlinear potential as compared to a parabolic potential (that results from a linear restoring force). In the experiment, random initial conditions arise from beam thermalization and nonuniform electron emission at the surface of the cathode. However, these characteristics alone do not explain the experimentally observed fluctuations in rf power and frequency. The predictions of the time- varying nonlinear potential theory clearly exhibits trends that were observed in the experimental results, in the form of classes of particle trajectories, fluctuations in particle asymptotic states, and particle motion sensitive to the shape of the virtual cathode. !10
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