Decomposition approach to exponential synchronisation for a class of non-linear singularly perturbed complex networks

This study is concerned with the globally exponential synchronisation problem for a class of non-linear singularly perturbed complex networks where all nodes possess the same structure and properties. The addressed global synchronisation problem is converted into the one for two lower-order sub-networks, namely, a non-linear slow sub-network and a linear fast sub-network, which are obtained by using the classical singular perturbation decomposition method. The network topology is directed and weighted, which means that the coupling configuration matrix is allowed to be asymmetric. By using the Lyapunov functional method and the Kronecker product technique, sufficient conditions are obtained under which the synchronisation is achieved, respectively, for the two sub-networks and the original complex network. These conditions can be easily verified by using the semi-definite programming method. A numerical example is finally simulated to validate the theoretical results and the effectiveness of the proposed synchronisation scheme.