Characterization and computation of partial synchronization manifolds for diffusive delay-coupled systems (Q2826316)

This paper deals with the study of partial synchronization of networks of dynamical systems. In contrast to full or no synchronization, partial synchronization is characterized by the fact that some systems in the network are synchronized identically while other are not. This phenomenon has also been observed in different branches of science.\newline \linebreak The networks under consideration are represented by directed weighted graphs and each of the finitely many nodes \(i\) of such a graph hosts a dynamical system of the form NEWLINE\[NEWLINE \begin{aligned}\dot{x}_i(t)&=f(x_i(t), u_i(t)),\\ y_i(t)&=h(x_i(t)), \end{aligned}\tag{1} NEWLINE\]NEWLINE where \(x_i(t)\in\mathbb{R}^n\) denotes the state, \(u_i(t)\in\mathbb{R}^{m}\) the input, and \(y_i(t)\in\mathbb{R}^m\) the output of the node \(i\) at time \(t\), and where \(f:\mathbb{R}^n\times\mathbb{R}^{m}\to\mathbb{R}^{n}\) and \(h:\mathbb{R}^n\to\mathbb{R}^m\) are two fixed functions satisfying some mild assumptions. The interaction of the systems (1) on the underlying graph is given by either one of the following linear coupling functions: NEWLINE\[NEWLINE u_i(t)=\sum_{j\in \mathcal{N}_i}a_{ij}[y_j(t-\tau)-y_i(t)],\tag{2} NEWLINE\]NEWLINE or NEWLINE\[NEWLINE u_i(t)=\sum_{j\in\mathcal{N}_i}a_{ij}[y_j(t-\tau)-y_i(t-\tau)].\tag{3} NEWLINE\]NEWLINE Here, the set \(\mathcal{N}_i\) stands for the neighbor set of the node \(i\) (i.e., \(\mathcal{N}_i\) is the set of all nodes \(j\) of the graph connected via an edge \((i,j)\) with the node \(i\)), the real number \(a_{ij}\geq 0\) defines the weight of the edge \((i,j)\), and the constant \(\tau\) denotes the delay.\newline \linebreak The authors associate partial synchronization of networks of systems (1) and coupling functions (2) or (3) with the existence of so-called partial synchronization manifolds, which form linear positively invariant subspaces of the state space of the network of systems. Then they prove three equivalent algebraic conditions -- compare Theorem 1 and Theorem 2 -- for the existence of partial synchronization manifolds: an invariant subspace condition, a row-sum condition, and a solvability condition for a particular matrix equation. Under additional mild assumptions, these conditions are not only sufficient for the existence of partial synchronization manifolds but also necessary as discussed in Theorem 3 and Theorem 4.NEWLINENEWLINEThe second part of the work addresses networks which are decomposable with respect to the rational dependency structure of the coupling weights, and with respect to the delay values, respectively. In the later case, the delay \(\tau\) involved in the coupling functions (2) or (3) is not constant but dependent on the indices \(i\) and \(j\); that is, the coupling functions contain multiple time-delays. Here, the authors show that if all subnetworks satisfy the existence conditions for partial synchronization manifolds simultaneously, then these conditions also hold for the original network. Moreover, they provide conditions where finding a partial synchronization manifold for the original network is equivalent to finding common partial synchronization manifolds for the underlying subnetsworks. Finally, the authors describe an available algorithm for detecting partial synchronization manifolds, and close with some final remarks.NEWLINENEWLINEOverall, the article is well written, contains some detailed examples demonstrating the obtained results, and includes some helpful notes and remarks related to the subject.