Synchronization in Discrete-Time, Discrete-State Random Dynamical Systems

摘要

We characterize the synchronization phenomenon in discrete-time, discrete-state random dynamical systems, with random and probabilistic Boolean networks as particular examples. In terms of multiplicative ergodic properties of the induced linear cocycle, we show such a random dynamical system with finite state synchronizes if and only if the Lyapunov exponent 0 has simple multiplicity. For the case of countable state set, characterization of synchronization is provided in term of the spectral subspace corresponding to the Lyapunov exponent -infinity. In addition, for cases of both finite and countable state sets, the mechanism of partial synchronization is described by partitioning the state set into synchronized subsets. Applications to biological networks are also discussed.