The strength of the direct calculation methodology for investigating the periodic solutions of the reverse flow reactor is demonstrated by studying two complex manifestations of the systems nonlinearity : multiplicity and instability. First we analyze the propane-propene fuel mixtures to determine all possible feed ranges where two stable, ignited solutions can co-exist. Calculations when catalytic sections of the reactor are replaced with inert sections show how this design modification can force the reactor to the desired stable solution. At other parameter values, we show how the Floquet stability analysis can be used in conjunction with direct calculation to detect bifurcations of symmetric solutions to unsymmetric and periodic solutions.