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For finite mode instabilities in dissipative systems, invariant manifold methods allow the bifurcation analysis to be reduced to the locally attracting center manifold. In a kinetic model of electron plasma dynamics, these methods are applied to the one mode beam-plasma instability which occurs via Hopf bifurcation. The instability results in a nonlinear oscillation, and the amplitude equation can be solved to describe the time asymptotic state.doi:10.1090/conm/028/751996 fatcat:r2nsx354pzeb5pjjb6w75d55gq