Itinerary synchronization in a network of nearly identical PWL systems coupled with unidirectional links and ring topology

TL;DR

This paper investigates the synchronization behavior of coupled piecewise linear systems arranged in a ring topology, introducing itinerary synchronization to detect multistability and collective dynamics influenced by the node with the fewest scrolls.

Contribution

It proposes the concept of itinerary synchronization for PWL systems, revealing how network topology and inner connection properties influence multistability and collective behavior.

Findings

Itinerary synchronization occurs when all state variables are coupled.

The node with the fewest scrolls determines the network's collective dynamics.

Network attacks can alter multistability and scroll preservation.

Abstract

In this paper the collective dynamics of $N$-coupled piecewise linear (PWL) systems with different number of scrolls and coupled in a master-slave sequence configuration is studied, i.e. a ring connection with unidirectional links. Itinerary synchronization is proposed to detect synchrony behavior with systems that can present generalized multistability. Itinerary synchronization consists in analyzing the symbolic dynamics of the systems by assigning different numbers to the regions where the scrolls are generated. It is shown that in certain parameter regimes if the inner connection between nodes is given by means of considering all the state variables of the system, 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 leading to generalized multistability phenomena which can be detected via itinerary synchronization. Results about attacks to the network are also presented, for example, when the PWL system is attacked by removing a given link to produce an open ring configuration. Depending on the inner connection properties, the nodes present multistability or preservation in the number of scrolls of the attractors.