Asymptotic behavior of some linear differential systems
type: Item
instance of: scholarly article
title: Asymptotic behavior of some linear differential systems
zbMATH Open document ID: 0909.34005
author: Tzuu-Shuh Chiang
author: Yunshyong Chow
published in: Journal of Differential Equations
publication date: 1999-04-12T00:00:00Z
review text: In the introduction, the authors consider a chasing game in \(\mathbb{R}\) among \(b\) particles whose trajectories are generated by the differential equations \[ F_i'(t)= \sum_{j\neq i} q_{ij}(t)(F_j(t)- F_i(t)) \] for \(i\in S= \{1,2,\dots,b\}\) where \(F_i(t)\) is the position of the particle \(i\) at time \(t\) and \(q_{ij}(t)\), which determines the instantaneous rate: the particle \(i\) is attracted toward particle \(j\), is \[ q_{ij}(t)= p_{ij} \lambda(t)^{U(i,j)}\quad\text{for }i\neq j. \] The rate function \(\lambda(t)> 0\) with \(\lim_{t\to\infty}\lambda(t)= 0\), \(U(i,j)\) is the ``cost'' function indicating of difficulty from \(i\) to \(j\) and the nonnegative number \(p_{ij}\) for \(i\neq j\) reflect the neighborhood structure of the particles. The authors suppose that the matrix \(P= (p_{ij})_{b\times b}\) is irreducible. A sufficient condition on \(\lambda(t)\) is given for all \(F_i(t)\) converging to a common finite limit (Theorem 1.1). In Section 2 some preliminary results are presented. Section 3 treats cycles of rank one and Section 4 deals with cycles of higher rank and a proof of Theorem 1.1. In the last part, some questions are presented for further study, besides proving the cycle structures for two typical ``cost'' functions.
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34E10
zbMATH DE Number: 1206604
zbMATH Keywords: linear
zbMATH Keywords: differential systems
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: cycle method
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3382
OpenAlex ID: W1981221197
cites work: Q4004079
cites work: A limit theorem for a class of inhomogeneous Markov processes
cites work: Q3672830
cites work: Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
cites work: Nonstationary Markov chains and convergence of the annealing algorithm
cites work: Cooling Schedules for Optimal Annealing
cites work: Singular perturbed Markov chains and exact behaviors of simulated annealing processes
cites work: Optimization by Simulated Annealing
cites work: Non-negative matrices and Markov chains. 2nd ed
cites work: Q3785827
cites work: Q4041702
cites work: Q5665539
DOI: 10.1006/JDEQ.1997.3382
description: scientific article; zbMATH DE number 1206604