Jalan Sarika, Jost Jürgen, Atay Fatihcan M
Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany.
Chaos. 2006 Sep;16(3):033124. doi: 10.1063/1.2336415.
We study coupled dynamics on networks using symbolic dynamics. The symbolic dynamics is defined by dividing the state space into a small number of regions (typically 2), and considering the relative frequencies of the transitions between those regions. It turns out that the global qualitative properties of the coupled dynamics can be classified into three different phases based on the synchronization of the variables and the homogeneity of the symbolic dynamics. Of particular interest is the homogeneous unsynchronized phase, where the coupled dynamics is in a chaotic unsynchronized state, but exhibits qualitative similar symbolic dynamics at all the nodes in the network. We refer to this dynamical behavior as symbolic synchronization. In this phase, the local symbolic dynamics of any arbitrarily selected node reflects global properties of the coupled dynamics, such as qualitative behavior of the largest Lyapunov exponent and phase synchronization. This phase depends mainly on the network architecture, and only to a smaller extent on the local chaotic dynamical function. We present results for two model dynamics, iterations of the one-dimensional logistic map and the two-dimensional Henon map, as local dynamical function.
我们使用符号动力学研究网络上的耦合动力学。符号动力学是通过将状态空间划分为少量区域（通常为2个）并考虑这些区域之间转移的相对频率来定义的。事实证明，基于变量的同步性和符号动力学的同质性，耦合动力学的全局定性特性可分为三个不同阶段。特别令人感兴趣的是均匀非同步阶段，其中耦合动力学处于混沌非同步状态，但在网络中的所有节点上都表现出定性相似的符号动力学。我们将这种动力学行为称为符号同步。在这个阶段，任意选择的节点的局部符号动力学反映了耦合动力学的全局特性，例如最大李雅普诺夫指数的定性行为和相位同步。这个阶段主要取决于网络架构，仅在较小程度上取决于局部混沌动力学函数。我们给出了两种模型动力学的结果，即作为局部动力学函数的一维逻辑斯谛映射和二维亨农映射的迭代。
