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We study coupled dynamics on networks using symbolic dynamics. The symbolicdynamics is defined by dividing the state space into a small number of regions(typically 2), and considering the relative frequencies of the transitionsbetween those regions. It turns out that the global qualitative properties ofthe coupled dynamics can be classified into three different phases based on thesynchronization of the variables and the homogeneity of the symbolic dynamics.Of particular interest is the {\it homogeneous unsynchronized phase} where thecoupled dynamics is in a chaotic unsynchronized state, but exhibits (almost)identical symbolic dynamics at all the nodes in the network. We refer to thisdynamical behaviour as {\it symbolic synchronization}. In this phase, the localsymbolic dynamics of any arbitrarily selected node reflects global propertiesof the coupled dynamics, such as qualitative behaviour of the largest Lyapunovexponent and phase synchronization. This phase depends mainly on the networkarchitecture, and only to a smaller extent on the local chaotic dynamicalfunction. We present results for two model dynamics, iterations of theone-dimensional logistic map and the two-dimensional H\'enon map, as localdynamical function.Comment: 21 pages, 7 figure

Symbolic synchronization and the detection of global properties of coupled dynamics from local information