Global and Partial Phase Synchronizations in Arrays of Piecewise Linear Time-Delay Systems

TL;DR

This paper investigates how arrays of piecewise linear time-delay systems achieve global or partial phase synchronization depending on boundary conditions and coupling strength, revealing sequential and cluster synchronization phenomena.

Contribution

It introduces the first detailed analysis of global and partial phase synchronization in arrays of piecewise linear time-delay systems with different boundary conditions.

Findings

Global phase synchronization occurs via sequential synchronization in open-ended arrays.

Partial phase synchronization forms stable clusters in ring topology arrays.

Both types of synchronization are confirmed through numerical methods.

Abstract

In this paper, we report the identification of global and partial phase synchronizations in linear arrays of unidirectionally coupled piecewise linear time-delay systems with two different coupling configurations. In particular, in a linear array with open end boundary conditions, global phase synchronization (GPS) is achieved by a sequential synchronization as a function of the coupling strength (a second order transition). Further, the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster. On the other hand, in a linear array with closed end boundary conditions (ring topology), partial phase synchronization (PPS) is achieved by forming different groups of phase synchronized clusters above some threshold value of the coupling strength (a first order transition) where they continue to be in a stable PPS state. We confirm the occurrence of both global and partial phase synchronizations in two different piecewise linear time-delay systems using various numerical methods.