Synchronization on star-like graphs and emerging $\mathbb{Z}_{p}$ symmetries at strong coupling

TL;DR

This paper investigates synchronization phenomena on inhomogeneous star-like graphs within the Kuramoto model, revealing discrete coupling values supporting states with emerging $ ext{Z}_p$ symmetries at strong coupling, supported by numerical and analytical methods.

Contribution

It introduces the analysis of synchronization transitions on long star graphs with inhomogeneity, identifying critical couplings and the emergence of $ ext{Z}_p$ symmetries at strong coupling, combining numerical and analytical approaches.

Findings

Different graph components synchronize at different critical couplings.

Discrete coupling values support $ ext{Z}_p$ symmetric states at strong coupling.

The stability of these states and their interpretation in Josephson arrays are discussed.

Abstract

We discuss the aspects of synchronization on inhomogeneous star-like graphs with long rays in Kuramoto model framework. We assume the positive correlation between internal frequencies and degrees for all nodes which supports the abrupt first order synchronization phase transition. It is found that different ingredients of the graph get synchronized at different critical couplings. Combining numerical and analytic tools we evaluate all critical couplings for the long star graph. Surprisingly it is found that at strong coupling there are discrete values of coupling constant which support the synchronized states with emerging $\mathbb{Z}_{p}$ symmetries. The stability of synchronized phase is discussed and the interpretation of phase with emerging $\mathbb{Z}_{p}$ symmetry for the Josephson array on long star graph is mentioned.