Global Complete Synchronization in Networks of Identical Stuart--Landau Oscillators

We study global complete synchronization in finite networks of diffusively coupled Stuart--Landau oscillators with identical natural frequencies. The underlying graph is arbitrary, connected, and undirected. In contrast with phase-only models, the amplitudes evolve dynamically and may approach zero, where the polar phase equations become singular. We derive explicit sufficient conditions that keep every amplitude uniformly separated from zero and then show that the classical invariant-arc mechanism for identical Kuramoto oscillators survives in this amplitude-inclusive setting. Two synchronization criteria are obtained, combining amplitude bounds, phase confinement, an energy identity, and the asymptotic structure of the synchronized manifold. Under either criterion, every oscillator converges exponentially to the same periodic orbit of radius $\sqrtμ$ and common frequency. The result provides a graph-level extension of the invariant-arc synchronization mechanism from identical Kuramoto networks to Stuart--Landau networks while retaining explicit control of the amplitude dynamics.