[en] Full text: A concern with ignited fusion plasmas is that, as a result of the instabilities they trigger, the high-energy particles eject themselves before they could give their energy to the core to sustain the reaction. Similarities between this class of instabilities and the so-called Berk-Breizman problem motivate us to study a single-mode instability driven by an energetic particle beam. For this purpose, a one dimensional Vlasov simulation is extended to include a Krook collision operator and external damping processes. The code is benchmarked with previous work. The fully nonlinear behavior is recovered in the whole parameter space characterized by an effective relaxation rate ν_a and an external damping rate γ_d. Steady state, periodic and chaotic behaviors are observed in nonlinear solutions. In the regime above marginal stability where both ν_a and γ_d are smaller than the linear drive γ_L, we observe a good agreement of steady saturation levels between the simulation and theory. Near marginal stability, the role of the normalized relaxation rate ν_a/(γ_L-γ_d), which is a key parameter to predict the behavior of the solution, is investigated for an initial distribution with relatively small γ_L, which correspond to the situation considered in the theory. In the low relaxation rate regime, frequency sweeping events are observed, and the time-evolution of such event is investigated. (author)