We now discuss briefly some limitations and extensions of the model considered in this MC 1568 work. Since we considered only the adiabatic case, we were able to use the invariants to simplify the analysis of the bifurcation features. While the adiabatic case provides some insight, heat losses as well as spatial gradients (and washcoat diffusion) are important and must be included in the modeling of real systems. The bifurcation analysis of such systems is obviously more complex but as stated earlier, all the features of the adiabatic lumped model will still be retained, and can serve as a starting point for the numerical investigation of the features of such systems. However, additional bifurcation phenomena may be observed, complicating the phase diagrams. Secondly, we used two modes to account for radial gradients, but if radial gradients are strong, a higher number of modes may be needed to describe the radial gradients in the gas temperature or within the washcoat. As we have already noted, tap root is possible that after homogeneous ignition, the gas phase temperature at the center of the channel may be larger than the wall temperature. Such solutions cannot be predicted using a 0-D 2-mode model and require at least three temperature modes (for which the natural choice is the average gas phase temperature, the cup-mixing average temperature and the solid phase temperature, see ) for the analysis. Such models as well as more detailed models with axial and radial gradients are topics for future investigations.