Reverse flow reactors are regarded as an appropriate tool for removing gaseous pollutants from industrial effluents. This paper presents the results of an experimental study on a particular reverse flow reactor with periodic variations in inlet concentration; these experimental results are compared with those obtained by numerical simulation studies. Catalytic oxidation of methane is chosen as model reaction. When the inlet concentration is low (i.e., dose to extinction limits) and periodically varied, the interaction between the feeding cycle period and the flow-reversal cycle period may lead to reactor instability; adjusting the period of flow reversal is one way to maintain thermal stability in such situations. Results of numerical investigations reveal that if the flow-reversal cycle time is an odd multiple of the inlet variation cycle time, a harmonic response will occur, with sharply increased variation of the maximum temperature. Basically, it is the inequality of total methane input during the two half-cycles during flow-reversal operations that cause reactor instability. Adjusting the phase between the two cycle periods may ease the inequality of the inlet concentration values, which procures a second strategy to enhance reactor stability. The temperature maximum is very sensitive to the heat-transfer parameters, when periodic inlet variations are introduced.