394 research outputs found
Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems
We provide Lyapunov-like characterizations of boundedness and convergence of
non-trivial solutions for a class of systems with unstable invariant sets.
Examples of systems to which the results may apply include interconnections of
stable subsystems with one-dimensional unstable dynamics or critically stable
dynamics. Systems of this type arise in problems of nonlinear output
regulation, parameter estimation and adaptive control.
In addition to providing boundedness and convergence criteria the results
allow to derive domains of initial conditions corresponding to solutions
leaving a given neighborhood of the origin at least once. In contrast to other
works addressing convergence issues in unstable systems, our results require
neither input-output characterizations for the stable part nor estimates of
convergence rates. The results are illustrated with examples, including the
analysis of phase synchronization of neural oscillators with heterogenous
coupling
Computational inference in systems biology
Parameter inference in mathematical models of biological pathways, expressed as coupled ordinary differential equations (ODEs), is a challenging problem. The computational costs associated with repeatedly solving the ODEs are often high. Aimed at reducing this cost, new concepts using gradient matching have been proposed. This paper combines current adaptive gradient matching approaches, using Gaussian processes, with a parallel tempering scheme, and conducts a comparative evaluation with current methods used for parameter inference in ODEs
Construction of an isotropic cellular automaton for a reaction-diffusion equation by means of a random walk
We propose a new method to construct an isotropic cellular automaton
corresponding to a reaction-diffusion equation. The method consists of
replacing the diffusion term and the reaction term of the reaction-diffusion
equation with a random walk of microscopic particles and a discrete vector
field which defines the time evolution of the particles. The cellular automaton
thus obtained can retain isotropy and therefore reproduces the patterns found
in the numerical solutions of the reaction-diffusion equation. As a specific
example, we apply the method to the Belousov-Zhabotinsky reaction in excitable
media
Structural Stability and Renormalization Group for Propagating Fronts
A solution to a given equation is structurally stable if it suffers only an
infinitesimal change when the equation (not the solution) is perturbed
infinitesimally. We have found that structural stability can be used as a
velocity selection principle for propagating fronts. We give examples, using
numerical and renormalization group methods.Comment: 14 pages, uiucmac.tex, no figure
Mapping degree in Banach spaces and spectral theory
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46287/1/209_2005_Article_BF01187933.pd
Conditions for propagation and block of excitation in an asymptotic model of atrial tissue
Detailed ionic models of cardiac cells are difficult for numerical
simulations because they consist of a large number of equations and contain
small parameters. The presence of small parameters, however, may be used for
asymptotic reduction of the models. Earlier results have shown that the
asymptotics of cardiac equations are non-standard. Here we apply such a novel
asymptotic method to an ionic model of human atrial tissue in order to obtain a
reduced but accurate model for the description of excitation fronts. Numerical
simulations of spiral waves in atrial tissue show that wave fronts of
propagating action potentials break-up and self-terminate. Our model, in
particular, yields a simple analytical criterion of propagation block, which is
similar in purpose but completely different in nature to the `Maxwell rule' in
the FitzHugh-Nagumo type models. Our new criterion agrees with direct numerical
simulations of break-up of re-entrant waves.Comment: Revised manuscript submitted to Biophysical Journal (30 pages incl.
10 figures
Observational operators for dual polarimetric radars in variational data assimilation systems (PolRad VAR v1.0)
We implemented two observational
operators for dual polarimetric radars in two variational data assimilation
systems: WRF Var, the Weather Research and Forecasting Model variational data
assimilation system, and NHM-4DVAR, the nonhydrostatic variational data
assimilation system for the Japan Meteorological Agency nonhydrostatic model.
The operators consist of a space interpolator, two types of variable
converters, and their linearized and transposed (adjoint) operators. The
space interpolator takes account of the effects of radar-beam broadening in
both the vertical and horizontal directions and climatological beam bending.
The first variable converter emulates polarimetric parameters with model
prognostic variables and includes attenuation effects, and the second one
derives rainwater content from the observed polarimetric parameter (specific
differential phase). We developed linearized and adjoint operators for the
space interpolator and variable converters and then assessed whether the
linearity of the linearized operators and the accuracy of the adjoint
operators were good enough for implementation in variational systems. The
results of a simple assimilation experiment showed good agreement between
assimilation results and observations with respect to reflectivity and
specific differential phase but not with respect to differential
reflectivity
A mathematical framework for critical transitions: normal forms, variance and applications
Critical transitions occur in a wide variety of applications including
mathematical biology, climate change, human physiology and economics. Therefore
it is highly desirable to find early-warning signs. We show that it is possible
to classify critical transitions by using bifurcation theory and normal forms
in the singular limit. Based on this elementary classification, we analyze
stochastic fluctuations and calculate scaling laws of the variance of
stochastic sample paths near critical transitions for fast subsystem
bifurcations up to codimension two. The theory is applied to several models:
the Stommel-Cessi box model for the thermohaline circulation from geoscience,
an epidemic-spreading model on an adaptive network, an activator-inhibitor
switch from systems biology, a predator-prey system from ecology and to the
Euler buckling problem from classical mechanics. For the Stommel-Cessi model we
compare different detrending techniques to calculate early-warning signs. In
the epidemics model we show that link densities could be better variables for
prediction than population densities. The activator-inhibitor switch
demonstrates effects in three time-scale systems and points out that excitable
cells and molecular units have information for subthreshold prediction. In the
predator-prey model explosive population growth near a codimension two
bifurcation is investigated and we show that early-warnings from normal forms
can be misleading in this context. In the biomechanical model we demonstrate
that early-warning signs for buckling depend crucially on the control strategy
near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio
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