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Forbidden sequences in symbolic dynamics of delayed maps and networks
Fatihcan M. Atay, Sarika Jalan and Jürgen Jost
We introduce the concept of avoiding sets for investigating symbolic dynamics of discrete dynamical systems with time delays. Such sets arise from specific non-generating partitions of the phase space that restrict the occurrence of particular symbol sequences related to the characteristics of the dynamics. We use the properties of avoiding sets in two applications to coupled map lattices, namely, determining unknown values of the transmission delays and detecting synchrony in networks of arbitrary topology with possibly directed and weighted connections. The method has the advantage of being based on a phase-space partition that is much easier to obtain than a generating partition; furthermore, it can utilize relatively short sequences of measurements from a single node of the network, it is independent of network size, and it performs well also in the presence of small noise.