The transient behavior of burner-supported spherical diffusion flames was studied in the transport-induced limit of low mass flow rate and the radiation-induced limit of high mass flow rate which characterize the isola response of flame extinction. Oscillatory instability was observed near both steady-state extinction limits. The oscillation typically grows in amplitude until it becomes large enough to extinguish the flame. The oscillatory behavior was numerically observed using detailed chemistry and transport for methane (50%CH4/50%He into 21%O-2/79%He) and hydrogen (100% H-2 into 21%O-2/79%He) diffusion flames where the fuel was issued from a point source, and helium was selected as an inert to increase the Lewis number, facilitating the onset of oscillation. In both methane and hydrogen flames, the oscillation always leads to extinction, and no limit cycle behavior was found. The growth rate of the oscillation was found to be slow enough under certain conditions to allow the flame to oscillate for over 450 s, suggesting that such oscillations can possibly be observed experimentally. For the hydrogen flames, however, the frequency of oscillation near the transport-induced limit is much larger, approximately 60 Hz as compared to 0.35 Hz for the methane flame, and the maximum amplitude of temperature oscillations was about 5 K. The distinctively different structures of the hydrogen and methane flames suggest that while both instabilities are thermal-diffusive in origin, oscillations in the hydrogen flames resemble those of premixed flames, while oscillations in the methane flames are non-premixed in character. (C) 2003 The Combustion Institute. All rights reserved.