Necessary and sufficient conditions for stable synchronization in random dynamical systems
Necessary and sufficient conditions for stable synchronization in random dynamical systems
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DOI：
10.1017/etds.2016.109
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发表时间：
2014-08
影响因子：
0.9
通讯作者：
J. Newman
中科院分区：
文献类型：
--
作者：
J. Newman
对于紧空间X上的独立同分布随机映射或无记忆随机流的合成，我们找到了存在局部渐近稳定的轨道(例如，由负的Lyapunov指数给出的)意味着任何给定的轨道对几乎必然相互收敛(同步)的条件。也就是说，我们发现当且仅当系统表现出以下性质时，同步发生并且是‘稳定的’：(I)存在一个最小的非空不变集$K\子集X$；(Ii)$K$中的任何两个点能够被移动到更近的位置；(Iii)$K$允许渐近稳定的轨迹。
For a composition of independent and identically distributed random maps or a memoryless stochastic flow on a compact space $X$ , we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies almost-sure mutual convergence of any given pair of trajectories (‘synchronization’). Namely, we find that synchronization occurs and is ‘stable’ if and only if the system exhibits the following properties: (i) there is a smallest non-empty invariant set $K\subset X$ ; (ii) any two points in $K$ are capable of being moved closer together; and (iii) $K$ admits asymptotically stable trajectories.