Department of Applied Mathematics, University of Waterloo
Existence and uniqueness of attracting slow manifolds: An application of the Ważewski principle
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 polyfacial set V for the system. The polyfacial set V must satisfy that its set of ingress points I are strict. The Ważewski Theorem asserts that there is a solution contained in V for all positive/negative values of time if the subset of strict ingress points of V is not a retract of the whole polyfacial set.
For the construction of the set V the existence of continuous 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 look at two things: 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.