A Vlasov code developed by Denavit [Phys. Fluids28, 2773 (1985)] was modified and used to study long‐time behavior of the single‐mode bump‐on‐tail instability. Large oscillations, which interrupted his original runs at &ohgr;0T≊1300 shortly after reaching the O’Neil trapping level (OTL), were identified and controlled by heating the cold background electrons. A run to &ohgr;0T=30 000 showed growth well beyond the OTL and eventually saturated at about the level predicted by Simon and Rosenbluth (SR) [Phys. Fluids19, 1567 (1976)], before oscillations set in. A run with warmer background electrons was again interrupted by oscillations. A weakly unstable case was followed to &ohgr;0T=60 000; it also showed growth well beyond the OTL but saturated at an amplitude that was significantly below that predicted by SR. In all cases, the velocity distribution developed strong ripples nearv=0. These are shown to be caused by a numerical grid effect, and they may be responsible for the eventual development of the large oscillations and for the low level of saturation in the weakly unstable case.