Slow manifold structure in explosive kinetics. 1. Bifurcations of points-at-infinity in prototypical models

This article analyzes in detail the global geometric properties (structure of the slow and fast manifolds) of prototypical models of explosive kinetics (the Semenov model for thermal explosion and the chain-branching model). The concepts of global or generalized slow manifolds and the notions of heterogeneity and alpha-omega inversion for invariant manifolds are introduced in order to classify the different geometric features exhibited by two-dimensional kinetic schemes by varying model parameters and to explain the phenomena that may occur in model reduction practice. This classification stems from the definition of suitable Lyapunov-type numbers and from the analysis of normal-to-tangent stretching rates. In the case of the Semenov model, we show that the existence of a global slow manifold and its properties are controlled by a transcritical bifurcation of the points-at-infinity, which can be readily identified by analyzing the Poincaré projected system. The issue of slow manifold uniqueness and the implications of the theory with regard to the practical definition of explosion limits are thoroughly addressed.