A systematic method is presented for constructing maps of parameter regions with qualitatively different bifurcation diagrams of the reverse-flow reactor (RFR). These maps are most useful for the rational design and periodic operation of an RFR. The method is illustrated by two examples. The map for a single, exothermic first-order reaction contains five regions with qualitatively different bifurcation diagrams when the Damkohler number is the bifurcation variable. The high-temperature branch is isolated in two of these regions. The map for two independent, exothermic first-order reactions contains seven regions with qualitatively different bifurcation diagrams when the adiabatic temperature rise is the bifurcation parameter. Three stable periodic stares exist for some values of the bifurcation variable in five of the regions. In some of these regions the intermediate state cannot be obtained upon a slow change in the concentration of the reactants. In all the examples we studied, a stability exchange of the pseudohomogeneous model occurred only at the limit points of the bifurcation diagrams. An efficient numerical scheme is presented for computing the loci of the singular points of the RFR model.