Close Returns Plots for Detecting a Chaotic Source in an Interaction Network

We are interested in studying the spread of chaos in an interaction network modeled by a Coupled Map Network (CMN). This graph is formed by nodes characterized by a measurable state variable that may exhibit chaotic time series. The interaction between the nodes may propagate their states in the network leading through a coupling process to some synchronization phenomenon which is known as nonlinear oscillator synchronization. The interaction network that we aim to study contains initially only one chaotic node that is responsible of the spread of chaos. Our goal consists then to study how to identify the node which is the source of the spread of chaos in an interaction network and how to detect the set of nodes becoming disturbed by the propagation of the chaotic node state. In this paper, we seek some appropriate measures to quantify the dynamic complexity of the nodes in order to identify the group of chaotic nodes as well as the source of the spread of chaos in the graph. We show by some simulations on random graphs that the Shannon entropy calculated on the close returns plots is an appropriate measure to detect chaotic series from a node. The extension of close returns plots to joint recurrence plots enables to identify the source of the spread of the disturbance in the network.