SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 233-251, January 2020.
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 $-\infty$. 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.

中文翻译：

---

离散时间，离散状态随机动力系统中的同步

SIAM应用动力系统杂志，第19卷，第1期，第233-251页，2020年1月。
我们以离散时间，离散状态随机动力系统为特征，以随机和概率布尔网络为例。就诱导线性共轭环的乘性遍历性质而言，我们表明，当且仅当Lyapunov指数0具有简单的多重性时，这种具有有限状态的随机动力学系统才会同步。对于可数状态集，根据对应于Lyapunov指数$-\ infty $的频谱子空间提供同步的表征。另外，对于有限状态集和可数状态集的情况，通过将状态集划分为同步子集来描述部分同步的机制。还讨论了对生物网络的应用。