Itinerary synchronization between PWL systems coupled with unidirectional links

Abstract In this paper the collective dynamics of N-coupled piecewise linear (PWL) systems with different number of scrolls is studied. The coupling is in a master-slave sequence configuration, with this type of coupling we investigate the synchrony behavior of a ring-connected network and a chain-connected network both with unidirectional links. Itinerary synchronization is used to detect synchrony behavior. Itinerary synchronization is defined in terms of the symbolic dynamics arising by assigning different numbers to the regions where the scrolls are generated. A weaker variant of this notion, ϵ-itinerary synchronization is also introduced and numerically investigated. It is shown that in certain parameter regimes if the inner connection between nodes takes account of all the state variables of the system (by which we mean that the inner coupling matrix is the identity matrix), then itinerary synchronization occurs and the coordinate motion is determined by the node with the smallest number of scrolls. Thus the collective behavior in all the nodes of the network is determined by the node with least scrolls in its attractor. Results about the dynamics in a directed chain topology are also presented. Depending on the inner connection properties, the nodes present multistability or preservation of the number of scrolls of the attractors.