Complete Synchronization and its Transition in Higher Harmonic Sakaguchi--Kuramoto Oscillators

TL;DR

This paper develops an analytical framework to induce complete synchronization in phase-frustrated higher harmonic Sakaguchi--Kuramoto oscillators, revealing the nature of transition types across different network topologies.

Contribution

It introduces optimal natural frequencies and coupling parameters to achieve synchronization, and analyzes transition types in various network models.

Findings

Complete synchronization can be induced at small coupling values.

Hysteresis indicates first-order transition in scale-free networks.

Second-order transition observed in Erdős–Rényi networks.

Abstract

In heterogeneous networks of coupled oscillators, phase frustration typically prevents the emergence of synchronization in the Sakaguchi--Kuramoto (SK) model. In this study, we propose an analytical framework to overcome this barrier and induce complete synchronization at a specified small coupling value in oscillators governed by phase-frustrated bi-harmonic coupling. We derive an optimal set of natural frequencies that is robust against added noise and correlated with the network degree heterogeneity, along with the parameters involved in the bi-harmonic coupling function that lead to complete synchronization ($r = 1$). In addition, we find complete synchronization transitions accompanied by hysteresis in scale-free networks, indicating a first-order (discontinuous) phase transition, whereas Erd\H{o}s--R\'enyi networks exhibit complete synchronization through a second-order (continuous) phase transition. Furthermore, we use the mean-field approximation in the presence of optimal frequencies to determine the critical coupling strength associated with the synchronization transition in the pure second-harmonic Sakaguchi--Kuramoto model. Here, the obtained optimal natural frequencies scale linearly with the node degree, and the critical coupling strength for the onset of synchronization is derived analytically from the self-consistent equations. In this specific regime, we observe a perfectly ordered two-cluster synchronized state. These findings remain robust for higher-order harmonic coupling schemes, as well as across a diverse range of synthetic and empirical networks, including scale-free, Erd\H{o}s--R\'enyi, Zachary Karate Club, and the \textit{C.~elegans} neural network, demonstrating their general applicability.