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 $g$ 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 $q$ 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