Directed Network Representation of Discrete Dynamical Maps

We suggest a network representation of dynamical maps (such as the logistic map) on the unit interval. The unit interval is partitioned into N subintervals, which are associated with ’nodes’ of the network. A link from node i to j is defined as a possible transition of the dynamical map from one interval i, to another j. In this way directed networks more generally allow phasespace representations (i.e. representations of transitions between different phasespace cells), of dynamical maps defined on finite intervals. We compute the diameter of these networks as well as the average path length between any two nodes. We numerically demonstrate that these network characteristics seem to diverge at the (first) zeros of the Lyapunov exponent and could thus provide an alternative measure to detect the ’edge of chaos’ in dynamical systems.