The fixed-bed reactor with periodic flow reversal has been shown to be a promising technology for the autothermal combustion of weakly concentrated gases. Modeling works have aided in building a better understanding of these systems, but using a dynamic simulation approach to reach the stable periodic orbit has been unsatisfactory. We propose a direct calculation method that transforms the initial-value problem into a steady-state boundary-value problem. In the steady-state formulation, standard bifurcation analysis techniques can be used to study the dependence of the system’s behavior on key parameters. Results for the catalytic combusion of trace amounts of propane in air illustrate the strengths of this method.