Asymptotic behavior of some linear differential systems (Q1265076)

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.