604 research outputs found
Oscillations in I/O monotone systems under negative feedback
Oscillatory behavior is a key property of many biological systems. The
Small-Gain Theorem (SGT) for input/output monotone systems provides a
sufficient condition for global asymptotic stability of an equilibrium and
hence its violation is a necessary condition for the existence of periodic
solutions. One advantage of the use of the monotone SGT technique is its
robustness with respect to all perturbations that preserve monotonicity and
stability properties of a very low-dimensional (in many interesting examples,
just one-dimensional) model reduction. This robustness makes the technique
useful in the analysis of molecular biological models in which there is large
uncertainty regarding the values of kinetic and other parameters. However,
verifying the conditions needed in order to apply the SGT is not always easy.
This paper provides an approach to the verification of the needed properties,
and illustrates the approach through an application to a classical model of
circadian oscillations, as a nontrivial ``case study,'' and also provides a
theorem in the converse direction of predicting oscillations when the SGT
conditions fail.Comment: Related work can be retrieved from second author's websit
Multi-Stability in Monotone Input/Output Systems
This paper studies the emergence of multi-stability and hysteresis in those
systems that arise, under positive feedback, starting from monotone systems
with well-defined steady-state responses. Such feedback configurations appear
routinely in several fields of application, and especially in biology.
Characterizations of global stability behavior are stated in terms of easily
checkable graphical conditions. An example of a signaling cascade under
positive feedback is presented.Comment: See http://www.math.rutgers.edu/~sontag for related work; to appear
in Systems and Control Letter
Monotone Control Systems
Monotone systems constitute one of the most important classes of dynamical
systems used in mathematical biology modeling.
The objective of this paper is to extend the notion of monotonicity to
systems with inputs and outputs, a necessary first step in trying to understand
interconnections, especially including feedback loops, built up out of monotone
components.
Basic definitions and theorems are provided, as well as an application to the
study of a model of one of the cell's most important subsystems.Comment: See http://www.math.rutgers.edu/~sontag/ for related wor
Convergence speed of unsteady distributed consensus: decay estimate along the settling spanning-trees
Results for estimating the convergence rate of non-stationary distributed
consensus algorithms are provided, on the basis of qualitative (mainly
topological) as well as basic quantitative information (lower-bounds on the
matrix entries). The results appear to be tight in a number of instances and
are illustrated through simple as well as more sophisticated examples. The main
idea is to follow propagation of information along certain spanning trees which
arise in the communication graph.Comment: 27 pages, 5 figure
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
- …