Existence and uniqueness of attracting slow manifolds: An application of the Ważewski principle
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Date
2017-01-23
Authors
Cid-Montiel, Lorena
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
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.
Description
Keywords
Ważewski principle, Retract method, Slow invariant manifolds