Periodic systems have new classes of synchronization stability

TL;DR

This paper extends the Master Stability Function framework to periodic systems, revealing new classes of synchronization stability and challenging assumptions about their ease of synchronization.

Contribution

It introduces a comprehensive classification of synchronization stability in periodic systems, identifying unique behaviors not present in chaotic systems.

Findings

Five classes of synchronization stability identified

Master Stability Function vanishes at the origin for periodic systems

Periodic systems can have higher thresholds for stable synchronization

Abstract

The Master Stability Function is a robust and useful tool for determining the conditions of synchronization stability in a network of coupled systems. While a comprehensive classification exists in the case in which the nodes are chaotic dynamical systems, its application to periodic systems has been less explored. By studying several well-known periodic systems, we establish a comprehensive framework to understand and classify their properties of synchronizability. This allows us to define five distinct classes of synchronization stability, including some that are unique to periodic systems. Specifically, in periodic systems, the Master Stability Function vanishes at the origin, and it can therefore display behavioral classes that are not achievable in chaotic systems, where it starts, instead, at a strictly positive value. Moreover, our results challenge the widely-held belief that periodic systems are easily put in a stable synchronous state, showing, instead, the common occurrence of a lower threshold for synchronization stability.