Existence and uniqueness of attracting slow manifolds: An application of the Ważewski principle

Abstract

In this work we present some of the geometric constructs that aid the application of the Ważewski Theorem. To illustrate the procedure the Michaelis-Menten mechanism will be considered. We show that M, a slow manifold, exists and is fully contained in a given set V . The set V must satisfy that its set of ingress points I with respect to de differential system of equations are strict. The Ważewski Theorem asserts that if the subset of strict ingress points of V is not a retract of the whole set then there exist a trajectory φ contained in V for all positive/negative values of time. More specifically, the theorem establishes that if we can find a set Z ⊂ V ∪ I such that Z ∩ I is a retract of I but not a retract of Z then φ exists. For the construction of the set V the existence of continuously differentiable functions which behave similarly to Liapunov functions on some parts of their zero-levels is required. The starting point to define such functions was to use the expressions obtained from the quasi steady state and rapid equilibrium assumptions (QSSA and REA). One surprising property of M is that it is the only trajectory that stays in the set V . To discuss uniqueness of the slow manifold we show the following two conditions are satisfied: • One of the coordinates, let us say xi is monotone and 0 < xi < ∞. The cross-section given by xi constant has either a non-decreasing or fixed diameter as xi increases. • The distance between two different solutions in V is non-decreasing as xi increases. with respect to a chosen variable, any two solutions in the polyfacial set V are always moving apart and the diameter of the cross sections of V is either decreasing or constant.