521 research outputs found
Subtropical Real Root Finding
We describe a new incomplete but terminating method for real root finding for
large multivariate polynomials. We take an abstract view of the polynomial as
the set of exponent vectors associated with sign information on the
coefficients. Then we employ linear programming to heuristically find roots.
There is a specialized variant for roots with exclusively positive coordinates,
which is of considerable interest for applications in chemistry and systems
biology. An implementation of our method combining the computer algebra system
Reduce with the linear programming solver Gurobi has been successfully applied
to input data originating from established mathematical models used in these
areas. We have solved several hundred problems with up to more than 800000
monomials in up to 10 variables with degrees up to 12. Our method has failed
due to its incompleteness in less than 8 percent of the cases
Towards an automated reduction method for polynomial ODE models in cellular biology
International audienceThis paper presents the first version of an algorithmic scheme dedicated to the model reduction problem, in the context of polynomial ODE models derived from generalized chemical reaction systems. This scheme, which relies on computer algebra, is implemented within a new MAPLE package. It is applied over an example. The qualitative analysis of the reduced model is afterwards completely carried out, proving the practical relevance of our methods
Combinatorial approaches to Hopf bifurcations in systems of interacting elements
We describe combinatorial approaches to the question of whether families of
real matrices admit pairs of nonreal eigenvalues passing through the imaginary
axis. When the matrices arise as Jacobian matrices in the study of dynamical
systems, these conditions provide necessary conditions for Hopf bifurcations to
occur in parameterised families of such systems. The techniques depend on the
spectral properties of additive compound matrices: in particular, we associate
with a product of matrices a signed, labelled digraph termed a DSR^[2] graph,
which encodes information about the second additive compound of this product. A
condition on the cycle structure of this digraph is shown to rule out the
possibility of nonreal eigenvalues with positive real part. The techniques
developed are applied to systems of interacting elements termed "interaction
networks", of which networks of chemical reactions are a special case.Comment: A number of minor errors and typos corrected, and some results
slightly improve
Algebraic Aspects of (Bio) Nano-chemical Reaction Networks and Bifurcations in Various Dynamical Systems
The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle configurations. We also investigate Hopf bifurcation surfaces of various dynamical systems
Better Answers to Real Questions
We consider existential problems over the reals. Extended quantifier
elimination generalizes the concept of regular quantifier elimination by
providing in addition answers, which are descriptions of possible assignments
for the quantified variables. Implementations of extended quantifier
elimination via virtual substitution have been successfully applied to various
problems in science and engineering. So far, the answers produced by these
implementations included infinitesimal and infinite numbers, which are hard to
interpret in practice. We introduce here a post-processing procedure to
convert, for fixed parameters, all answers into standard real numbers. The
relevance of our procedure is demonstrated by application of our implementation
to various examples from the literature, where it significantly improves the
quality of the results
Chimera states in networks of phase oscillators: the case of two small populations
Chimera states are dynamical patterns in networks of coupled oscillators in
which regions of synchronous and asynchronous oscillation coexist. Although
these states are typically observed in large ensembles of oscillators and
analyzed in the continuum limit, chimeras may also occur in systems with finite
(and small) numbers of oscillators. Focusing on networks of phase
oscillators that are organized in two groups, we find that chimera states,
corresponding to attracting periodic orbits, appear with as few as two
oscillators per group and demonstrate that for the bifurcations that
create them are analogous to those observed in the continuum limit. These
findings suggest that chimeras, which bear striking similarities to dynamical
patterns in nature, are observable and robust in small networks that are
relevant to a variety of real-world systems.Comment: 13 pages, 16 figure
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