Hopf bifurcation and the beam-plasma instability

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