Synchronization and Hopf Bifurcation in Stuart--Landau Networks

TL;DR

This paper explores how amplitude dynamics influence synchronization in networks of Stuart--Landau oscillators, revealing conditions for stable synchronization and the emergence of collective oscillations near Hopf bifurcations.

Contribution

It extends classical Kuramoto models by incorporating amplitude dynamics and analyzes synchronization and bifurcation phenomena in Stuart--Landau networks.

Findings

Established topology-robust complete synchronization conditions

Demonstrated emergence of synchronized periodic states at Hopf bifurcation

Clarified the role of amplitude dynamics in network synchronization

Abstract

The Kuramoto model has shaped our understanding of synchronization in complex systems, yet its phase-only formulation neglects amplitude dynamics that are intrinsic to many oscillatory networks. In this work, we revisit Kuramoto-type synchronization through networks of Stuart--Landau oscillators, which arise as the universal normal form near a Hopf bifurcation. For identical natural frequencies, we analyze synchronization in two complementary regimes. Away from criticality, we establish topology-robust complete synchronization for general connected networks under explicit sufficient conditions that preclude amplitude death. At criticality, we exploit network symmetries to analyze the onset of collective oscillations via Hopf bifurcation theory, demonstrating the emergence of synchronized periodic states in ring-symmetric networks. Our results clarify how amplitude dynamics enrich the structure of synchronized states and provide a bridge between classical Kuramoto synchronization and amplitude-inclusive models in complex networks.