Synchronization for complex networks with Markov switching via matrix measure approach

This paper devotes to almost sure synchronization and almost sure quasi-synchronization of complex networks with Markov switching. Some sufficient conditions are derived in terms of the ergodic theory of continuous time Markov chain and the matrix measure approach, which can guarantee that the dynamical networks almost surely synchronize or quasi-synchronize to a given manifold. According to the property of Markov chain and the exponential distribution of switching time sequence, we also estimate the probability distribution of the quasi-synchronization error for a two-state Markov chain and then generalize them to a finite state space Markov chain. Meanwhile, some examples with numerical simulations are given to show that the Markov chain plays an important role in synchronization of networks.