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arXiv · 2601.10234

Synchronization and Hopf Bifurcation in Stuart--Landau Networks

Kuan-Wei Chen·Ting-Yang Hsiao
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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 exponential complete synchronization on arbitrary finite connected undirected networks under explicit sufficient conditions on the parameters and initial data that prevent amplitude death. For ring networks, we identify an exact branch of synchronous periodic solutions arising from a supercritical Hopf bifurcation and use block-circulant Fourier analysis to determine the critical parameter values and multiplicities of the non-synchronous modes. For $N=7$ and $s=2$, a center-manifold reduction yields cubic amplitude equations for paired critical modes, identifying exact single-mode rotating-wave solutions and a standing-wave pattern at cubic order. Numerical simulations compare the dynamics restricted to the rotating-wave invariant subspaces with direct simulations of the full network.

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Kuan-Wei Chen, Ting-Yang Hsiao. 2026-01-15. Synchronization and Hopf Bifurcation in Stuart--Landau Networks. https://arxiv.org/abs/2601.10234

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