69 research outputs found
Graph-theoretic conditions for injectivity of functions on rectangular domains
This paper presents sufficient graph-theoretic conditions for injectivity of
collections of differentiable functions on rectangular subsets of R^n. The
results have implications for the possibility of multiple fixed points of maps
and flows. Well-known results on systems with signed Jacobians are shown to be
easy corollaries of more general results presented here.Comment: 16 pages, 5 figure
Global convergence in systems of differential equations arising from chemical reaction networks
It is shown that certain classes of differential equations arising from the
modelling of chemical reaction networks have the following property: the state
space is foliated by invariant subspaces each of which contains a unique
equilibrium which, in turn, attracts all initial conditions on the associated
subspace.Comment: Some typos and minor errors from the previous version have been
correcte
Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements
We extend previous work on injectivity in chemical reaction networks to
general interaction networks. Matrix- and graph-theoretic conditions for
injectivity of these systems are presented. A particular signed, directed,
labelled, bipartite multigraph, termed the ``DSR graph'', is shown to be a
useful representation of an interaction network when discussing questions of
injectivity. A graph-theoretic condition, developed previously in the context
of chemical reaction networks, is shown to be sufficient to guarantee
injectivity for a large class of systems. The graph-theoretic condition is
simple to state and often easy to check. Examples are presented to illustrate
the wide applicability of the theory developed.Comment: 34 pages, minor corrections and clarifications on previous versio
Building oscillatory chemical reaction networks by adding reversible reactions
We show that if a chemical reaction network (CRN) admits nondegenerate
(resp., linearly stable) oscillation, and we add new reversible reactions
involving new species to this CRN, then the new CRN so created also admits
nondegenerate (resp., linearly stable) oscillation provided certain mild and
easily checked conditions are met. This claim that the larger CRN "inherits"
oscillation from the smaller one, provided it is built from the smaller CRN in
an appropriate way, follows an analogous result involving multistationarity. It
also adds to a number of prior results on the inheritance of oscillation; these
collectively often allow us to determine the capacity of a given network for
oscillation based on an analysis of its subnetworks.Comment: updated to elucidate the proo
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