Emergent Behaviors in Coupled Multi-scroll Oscillators in Network with Subnetworks

Abstract

This paper presents the emergence of two collective behaviors in interconnected networks. Specifically, the nodes in these networks belong to a particular class of piece-wise linear systems. The global topology of the network is designed in the form of connected subnetworks, which do not necessarily share the same structure and coupling strength. In particular, it is considered that there are two levels of connection, the internal level is related to the connection between the nodes of each subnetwork; while the external level is related to connections between subnetworks. In this configuration, the internal level is considered to provide lower bounds on the coupling strength to ensure internal synchronization of subnetworks. The external level has a relevant value in the type of collective behavior that can be achieved, for which, we determine conditions in the coupling scheme, to achieve partial or complete cluster synchronization, preserving the internal synchronization of each cluster. The analysis of the emergence of stable collective behavior is presented by using Lyapunov functions of the different coupling. The theoretical results are validated by numerical simulations.