Changes in the qualitative features of the bifurcation diagrams or the dynamic features of forced periodic systems occur at singular points, which satisfy certain defining conditions. The loci of these singular points may be constructed by a continuation procedure and used to bound parameter regions with qualitatively different features. When the model of a forced periodic system is a set of partial differential equations, construction of these loci may require extensive computational time, making this task often impractical. We present here a novel, very efficient numerical method for construction of these loci. The procedure uses Frechet differentiation to simplify the determination of the defining conditions and the Broyden inverse update method to accelerate the iterative steps involved in the shooting method. The procedure is illustrated first by construction of a map of parameter regions with qualitatively different bifurcation diagrams for an adiabatic reverse-flow reactor (RFR), the direction of feed to which is changed periodically. We then construct a map of parameter regions in which a cooled RFR has qualitatively different dynamic features. Both maps reveal surprising features.