33,866 research outputs found
Differential Inequalities in Multi-Agent Coordination and Opinion Dynamics Modeling
Distributed algorithms of multi-agent coordination have attracted substantial
attention from the research community; the simplest and most thoroughly studied
of them are consensus protocols in the form of differential or difference
equations over general time-varying weighted graphs. These graphs are usually
characterized algebraically by their associated Laplacian matrices. Network
algorithms with similar algebraic graph theoretic structures, called being of
Laplacian-type in this paper, also arise in other related multi-agent control
problems, such as aggregation and containment control, target surrounding,
distributed optimization and modeling of opinion evolution in social groups. In
spite of their similarities, each of such algorithms has often been studied
using separate mathematical techniques. In this paper, a novel approach is
offered, allowing a unified and elegant way to examine many Laplacian-type
algorithms for multi-agent coordination. This approach is based on the analysis
of some differential or difference inequalities that have to be satisfied by
the some "outputs" of the agents (e.g. the distances to the desired set in
aggregation problems). Although such inequalities may have many unbounded
solutions, under natural graphic connectivity conditions all their bounded
solutions converge (and even reach consensus), entailing the convergence of the
corresponding distributed algorithms. In the theory of differential equations
the absence of bounded non-convergent solutions is referred to as the
equation's dichotomy. In this paper, we establish the dichotomy criteria of
Laplacian-type differential and difference inequalities and show that these
criteria enable one to extend a number of recent results, concerned with
Laplacian-type algorithms for multi-agent coordination and modeling opinion
formation in social groups.Comment: accepted to Automatic
Opinion Dynamics in Social Networks with Hostile Camps: Consensus vs. Polarization
Most of the distributed protocols for multi-agent consensus assume that the
agents are mutually cooperative and "trustful," and so the couplings among the
agents bring the values of their states closer. Opinion dynamics in social
groups, however, require beyond these conventional models due to ubiquitous
competition and distrust between some pairs of agents, which are usually
characterized by repulsive couplings and may lead to clustering of the
opinions. A simple yet insightful model of opinion dynamics with both
attractive and repulsive couplings was proposed recently by C. Altafini, who
examined first-order consensus algorithms over static signed graphs. This
protocol establishes modulus consensus, where the opinions become the same in
modulus but may differ in signs. In this paper, we extend the modulus consensus
model to the case where the network topology is an arbitrary time-varying
signed graph and prove reaching modulus consensus under mild sufficient
conditions of uniform connectivity of the graph. For cut-balanced graphs, not
only sufficient, but also necessary conditions for modulus consensus are given.Comment: scheduled for publication in IEEE Transactions on Automatic Control,
2016, vol. 61, no. 7 (accepted in August 2015
Uniform attractors for non-autonomous wave equations with nonlinear damping
We consider dynamical behavior of non-autonomous wave-type evolutionary
equations with nonlinear damping, critical nonlinearity, and time-dependent
external forcing which is translation bounded but not translation compact
(i.e., external forcing is not necessarily time-periodic, quasi-periodic or
almost periodic). A sufficient and necessary condition for the existence of
uniform attractors is established using the concept of uniform asymptotic
compactness. The required compactness for the existence of uniform attractors
is then fulfilled by some new a priori estimates for concrete wave type
equations arising from applications. The structure of uniform attractors is
obtained by constructing a skew product flow on the extended phase space for
the norm-to-weak continuous process.Comment: 33 pages, no figur
Shadowing by non uniformly hyperbolic periodic points and uniform hyperbolicity
We prove that, under a mild condition on the hyperbolicity of its periodic
points, a map which is topologically conjugated to a hyperbolic map
(respectively, an expanding map) is also a hyperbolic map (respectively, an
expanding map). In particular, this result gives a partial positive answer for
a question done by A. Katok, in a related context
Erraticity of Rapidity Gaps
The use of rapidity gaps is proposed as a measure of the spatial pattern of
an event. When the event multiplicity is low, the gaps between neighboring
particles carry far more information about an event than multiplicity spikes,
which may occur very rarely. Two moments of the gap distrubiton are suggested
for characterizing an event. The fluctuations of those moments from event to
event are then quantified by an entropy-like measure, which serves to describe
erraticity. We use ECOMB to simulate the exclusive rapidity distribution of
each event, from which the erraticity measures are calculated. The dependences
of those measures on the order of of the moments provide single-parameter
characterizations of erraticity.Comment: 10 pages LaTeX + 5 figures p
Freezing of Nonlinear Bloch Oscillations in the Generalized Discrete Nonlinear Schrodinger Equation
The dynamics in a nonlinear Schrodinger chain in an homogeneous electric
field is studied. We show that discrete translational invariant
integrability-breaking terms can freeze the Bloch nonlinear oscillations and
introduce new faster frequencies in their dynamics. These phenomena are studied
by direct numerical integration and through an adiabatic approximation. The
adiabatic approximation allows a description in terms of an effective potential
that greatly clarifies the phenomenon.Comment: LaTeX, 7 pages, 6 figures. Improved version to appear in Phys. Rev.
Spectral analysis of electron transfer kinetics II
Electron transfer processes in Debye solvents are studied using a spectral
analysis method recently proposed. Spectral structure of a nonadiabatic
two-state diffusion equation is investigated to reveal various kinetic regimes
characterized by a broad range of physical parameters; electronic coupling,
energy bias, reorganization energy, and solvent relaxation rate. Within this
unified framework, several kinetic behaviors of the electron transfer kinetics,
including adiabatic Rabi oscillation, crossover from the nonadiabatic to
adiabatic limits, transition from the incoherent to coherent kinetic limits,
and dynamic bath effect, are demonstrated and compared with results from
previous theoretical models. Dynamics of the electron transfer system is also
calculated with the spectral analysis method. It is pointed out that in the
large reorganization energy case the nonadiabatic diffusion equation exhibits a
non-physical behavior, yielding a negative eigenvalue.Comment: submitted for a publication in Journal of Chemical Physic
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