Generalized connection graph method for synchronization in asymmetrical networks (Q858505)

The paper considers the network of identical oscillators \[ x_i'=F(x_i) + \sum_{k=1}^{n} d_{ik}(t)Px_k, \quad i=1,\dots,n, \] where \(x_i\in \mathbb{R}^l\), \(F: \mathbb{R}^l\to \mathbb{R}^l\), \(P\) is a projection matrix that selects the components of \(x_i\) that are involved in the coupling. The connection matrix \(D=\{d_{ij}(t)\}\) is assumed to satisfy \[ \sum_k d_{ik} = 0,\quad d_{ii}=-\sum_{k\neq i} d_{ik},\quad i=1,\dots,n. \] The main result provides conditions for global complete synchronization of this network, i.e. \(\| x_i(t)-x_{j}(t)\| \to 0\) for all \(i,j\) and \(t\to \infty\). These conditions extend the results of the authors in [Physica D 195, No. 1--2, 159--187 (2004; Zbl 1098.82622)] to the case of arbitrary asymmetrical connection matrix \(D\).