A one-step reduced chemical-kinetic mechanism describing near-critical hydrogen combustion, recently derived by assuming that all chemical intermediates maintain steady state, is used to investigate numerically the pulsating dynamics of fuel-rich hydrogen-air flames. The computations, considering pressures of up to 20 atmospheres, address the flame evolution for increasing values of the equivalence ratio ϕ, a relevant bifurcation parameter. Besides critical conditions associated with the Hopf bifurcation occurring at a critical value ϕc of ϕ, supercritical dynamics for ϕ>ϕc is investigated, including the nonlinear gradual growth of the oscillation amplitude with increasing ϕ, the occurrence of a period-doubling bifurcation, and the emergence of nonlinear relaxation oscillations. Comparisons with results of numerical calculations employing detailed chemistry reveal that the one-step description is able to predict with unexpectedly good accuracy the flame dynamics, including the critical conditions at the bifurcations as well as the nonlinear dynamics encountered for ϕ>ϕc.