763 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
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
Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach
Motivated by the result of invariance of regular-boundary open sets in
\cite{CannarsaDaPratoFrankowska2009} and multi-stability issues in gene
networks, our paper focuses on three closely related aims. First, we give a
necessary local Lipschitz-like condition in order to expect invariance of open
sets (for deterministic systems). Comments on optimality are provided via
examples. Second, we provide a border avoidance (near-viability) counterpart of
\cite{CannarsaDaPratoFrankowska2009} for controlled Brownian diffusions and
piecewise deterministic switched Markov processes (PDsMP). We equally discuss
to which extent Lipschitz-continuity of the driving coefficients is needed.
Finally, by applying the theoretical result on PDsMP to Hasty's model of
bacteriophage (\cite{hasty\_pradines\_dolnik\_collins\_00},
\cite{crudu\_debussche\_radulescu\_09}), we show the necessity of explicit
modeling for the environmental cue triggering lysis
Unstable Dynamics, Nonequilibrium Phases and Criticality in Networked Excitable Media
Here we numerically study a model of excitable media, namely, a network with
occasionally quiet nodes and connection weights that vary with activity on a
short-time scale. Even in the absence of stimuli, this exhibits unstable
dynamics, nonequilibrium phases -including one in which the global activity
wanders irregularly among attractors- and 1/f noise while the system falls into
the most irregular behavior. A net result is resilience which results in an
efficient search in the model attractors space that can explain the origin of
certain phenomenology in neural, genetic and ill-condensed matter systems. By
extensive computer simulation we also address a relation previously conjectured
between observed power-law distributions and the occurrence of a "critical
state" during functionality of (e.g.) cortical networks, and describe the
precise nature of such criticality in the model.Comment: 18 pages, 9 figure
Constructing solutions to the Bj\"orling problem for isothermic surfaces by structure preserving discretization
In this article, we study an analog of the Bj\"orling problem for isothermic
surfaces (that are more general than minimal surfaces): given a real analytic
curve in , and two analytic non-vanishing orthogonal
vector fields and along , find an isothermic surface that is
tangent to and that has and as principal directions of
curvature. We prove that solutions to that problem can be obtained by
constructing a family of discrete isothermic surfaces (in the sense of Bobenko
and Pinkall) from data that is sampled along , and passing to the limit
of vanishing mesh size. The proof relies on a rephrasing of the
Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its
discretization which is induced from the geometry of discrete isothermic
surfaces. The discrete-to-continuous limit is carried out for the Christoffel
and the Darboux transformations as well.Comment: 29 pages, some figure
Invariance Conditions for Nonlinear Dynamical Systems
Recently, Horv\'ath, Song, and Terlaky [\emph{A novel unified approach to
invariance condition of dynamical system, submitted to Applied Mathematics and
Computation}] proposed a novel unified approach to study, i.e., invariance
conditions, sufficient and necessary conditions, under which some convex sets
are invariant sets for linear dynamical systems.
In this paper, by utilizing analogous methodology, we generalize the results
for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the
nonlinear Farkas lemma and the \emph{S}-lemma, together with Nagumo's Theorem
are utilized to derive invariance conditions for discrete and continuous
systems. Only standard assumptions are needed to establish invariance of
broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we
establish an optimization framework to computationally verify the derived
invariance conditions. Finally, we derive analogous invariance conditions
without any conditions
Mapping degree in Banach spaces and spectral theory
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46287/1/209_2005_Article_BF01187933.pd
Noise Induced Coherence in Neural Networks
We investigate numerically the dynamics of large networks of globally
pulse-coupled integrate and fire neurons in a noise-induced synchronized state.
The powerspectrum of an individual element within the network is shown to
exhibit in the thermodynamic limit () a broadband peak and an
additional delta-function peak that is absent from the powerspectrum of an
isolated element. The powerspectrum of the mean output signal only exhibits the
delta-function peak. These results are explained analytically in an exactly
soluble oscillator model with global phase coupling.Comment: 4 pages ReVTeX and 3 postscript figure
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