Synchronization analysis of coupled differential systems with time-varying couplings

This paper considers the problem of synchronization of differential dynamical systems with time-varying coupling. The temporal variation of the couplings we consider here is rather general and includes variations in both the network structure and the reaction dynamics. For example, driven by a metric dynamical system. Inspired by the author's previous work [1], the generalized Hajnal diameter is introduced to study the stability of synchronization manifold via the underlying variational equation, which can be proved to equal to e^λ, where λ is the largest one of all Lyapunov exponents of the underlying variational equation, corresponding to the space transverses to the synchronization manifold. As an application, these results are used to investigate the synchronization of linearly coupled ordinary differential systems (LCODEs) with identity inner coupling matrix. In this case, the Hajnal diameter of the linear system induced by the time-varying coupling matrices can be used to measure the synchronizability of the time-varying coupling process. The corresponding network can synchronize some chaotic attractors if and only if there exists some T > 0 such that the joint union of the topologies across any T-length time interval has spanning trees.