The fixed-bed reactor with periodical flow reversal allows a stable autothermal operating condition, even for a weak exothermal reaction. Periodical flow reversal has also been proposed for the combustion of gases with traces of N2O. It is a fact that higher temperatures can lead to a hybrid reaction, i.e. in addition to the heterogeneous-catalytic reaction a homogeneous-thermal combustion takes place. The following investigates the influence of axially structured monolith catalyst on the hybrid N2O decomposition in order to understand the coupling mechanisms in more detail. Recent studies prove the potential of the direct calculation of cyclic-steady states which transform the usual initial-value problem into a steady-state boundary-value problem. In combining this technique with standard bifurcation analysis tools, it is now possible to study the influence of key parameters on the process in a very e cient and elegant way. Both, simulations and experimental investigations were required to design an appropriate setup for a spatial and temporal separation of the two underlying types of reaction.