Synchronization and Hopf Bifurcation in Stuart--Landau Networks

TL;DR

By moving beyond phase-only models, the paper studies synchronization in networks of Stuart--Landau oscillators with diffusive coupling. It develops a diffusive model $\dot z_j=(\mu+i\omega_j)z_j-|z_j|^2z_j+c\sum_k a_{jk}(z_k-z_j)$, proves topology-robust exponential frequency-amplitude synchronization away from criticality for identical frequencies, and analyzes Hopf bifurcation at $\mu=0$ on ring networks to produce synchronous periodic states. The analysis yields explicit anti-amplitude-death conditions, energy-dissipation arguments on general graphs, and a block-circulant spectral decomposition with eigenvalues $\lambda_j^{\pm}(\mu)=(\mu-\mu_j)\pm i\omega$ for non-synchronous modes; all-to-all coupling yields degenerate spectra. These results bridge Kuramoto-type phase synchronization and amplitude-inclusive dynamics, with implications for engineering and understanding coherence in complex oscillator networks.

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.

Figures (2)

• Figure 1: Bifurcation diagrams of system \ref{['SL-network-xy-N']} with $\omega=1$ and $c=0.05$ for $N=6$ and $s=2$. The horizontal axis is the bifurcation parameter $\mu$, and the vertical axis shows the maxima of $x_1$ (left) and $x_2$ (right) along the computed branches. A simple Hopf bifurcation occurs at $\mu=0$ (point 2). Additional critical parameter values are $\mu_2=\mu_6=0.2$ (point 3) and $\mu_3=\mu_5=0.3$ (point 4); the unpaired index is $j^\ast=4$ with $\mu_4=0.2$ (point 3).
• Figure 2: Bifurcation diagrams of system \ref{['SL-network-xy-N']} with $\omega=1$ and $c=0.05$ for $N=7$ and $s=2$. A simple Hopf bifurcation occurs at $\mu=0$ (point 2). Additional critical parameter values satisfy $\mu_2=\mu_7\approx 0.159903$ (point 3), $\mu_4=\mu_5\approx 0.22775$ (point 4), and $\mu_3=\mu_6\approx 0.31235$ (point 5). In this odd-$N$ case, all additional critical values correspond to paired modes, and hence none of the associated Hopf critical points is simple.

Theorems & Definitions (31)

• Remark 2.1
• Remark 2.2: Phases and unwrapping
• Remark 2.3
• Remark 2.4
• Definition 2.1: Frequency-amplitude synchronization
• Definition 2.2: Complete synchronization
• Remark 2.5
• Lemma 2.1: Instability of Complete Amplitude Death
• proof
• Lemma 2.2: Persistence of Nonvanishing Amplitudes