A simple one-field L-H transition model is studied in detail, analytically and numerically. The dynamical system consists of three equations coupling the drift wave turbulence level, zonal flow speed, and the pressure gradient. The fourth component, i.e., the mean shear velocity, is slaved to the pressure gradient. Bursting behavior, characteristic for predator-prey models of the drift wave - zonal flow interaction, is recovered near the transition to the quiescent H-mode (QH) and occurs as strongly nonlinear relaxation oscillations. The latter, in turn, arise as a result of Hopf bifurcation (limit cycle) of an intermediate fixed point (between the L- and H-modes). The system is shown to remain at the QH-mode fixed point even after the heating rate is decreased below the bifurcation point (i.e., hysteresis, subcritical bifurcation), but the basin of attraction of the QH-mode shrinks rapidly with decreasing power. This suggests that the hysteresis in the H-L transition may be less than that expected from S-curve models. Nevertheless, it is demonstrated that by shaping the heating rate temporal profile, one can reduce the average power required for the transition to the QH-mode. © 2009 American Institute of Physics.