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Synchronization of coupled Stuart-Landau oscillators: How heterogeneity can facilitate synchronization

Ana P Millán, David Poyato, David N Reynolds, Francesco Tudisco

TL;DR

This work analyzes a finite network of Stuart–Landau oscillators with heterogeneous Hopf parameters and natural frequencies, focusing on the full amplitude–phase dynamics beyond phase reductions. By decomposing into amplitudes and phases, it uncovers phenomena absent in Kuramoto-type models, such as leader-driven synchronization and amplitude-death transitions, and provides a complete asymptotic classification for the two-oscillator case. For general $N$ with zero frequency heterogeneity, it proves exponential phase synchronization under sectorial initial data and describes the limiting amplitude structure, including bounds when oscillators remain active. A real-valued reduction reveals a nonlinear opinion-dynamics interpretation with fixed points corresponding to disagreement, compromise, and consensus, enriching the connections between synchronization and collective decision processes.

Abstract

We study the collective dynamics of coupled Stuart--Landau oscillators, which model limit-cycle behavior near a Hopf bifurcation and serve as the amplitude-phase analogue of the Kuramoto model. Unlike the well-studied phase-reduced systems, the full Stuart--Landau model retains amplitude dynamics, enabling the emergence of rich phenomena such as amplitude death, quenching, and multistable synchronization. We provide a complete analytical classification of asymptotic behaviors for identical natural frequencies, but heterogeneous inherent amplitudes in the finite-$N$ setting. In the two-oscillator case, we classify the asymptotic behavior in all possible regimes including heterogeneous natural frequencies and inherent amplitudes, and in particular we identify and characterize a novel regime of \emph{leader-driven synchronization}, wherein one active oscillator can entrain another regardless of frequency mismatch. For general $N$, we prove exponential phase synchronization under sectorial initial data and establish sharp conditions for global amplitude death. Finally, we analyze a real-valued reduction of the model, connecting the dynamics to nonlinear opinion formation and consensus processes. Our results highlight the fundamental differences between amplitude-phase and phase-only Kuramoto models, and provide a new framework for understanding synchronization in heterogeneous oscillator networks.

Synchronization of coupled Stuart-Landau oscillators: How heterogeneity can facilitate synchronization

TL;DR

This work analyzes a finite network of Stuart–Landau oscillators with heterogeneous Hopf parameters and natural frequencies, focusing on the full amplitude–phase dynamics beyond phase reductions. By decomposing into amplitudes and phases, it uncovers phenomena absent in Kuramoto-type models, such as leader-driven synchronization and amplitude-death transitions, and provides a complete asymptotic classification for the two-oscillator case. For general with zero frequency heterogeneity, it proves exponential phase synchronization under sectorial initial data and describes the limiting amplitude structure, including bounds when oscillators remain active. A real-valued reduction reveals a nonlinear opinion-dynamics interpretation with fixed points corresponding to disagreement, compromise, and consensus, enriching the connections between synchronization and collective decision processes.

Abstract

We study the collective dynamics of coupled Stuart--Landau oscillators, which model limit-cycle behavior near a Hopf bifurcation and serve as the amplitude-phase analogue of the Kuramoto model. Unlike the well-studied phase-reduced systems, the full Stuart--Landau model retains amplitude dynamics, enabling the emergence of rich phenomena such as amplitude death, quenching, and multistable synchronization. We provide a complete analytical classification of asymptotic behaviors for identical natural frequencies, but heterogeneous inherent amplitudes in the finite- setting. In the two-oscillator case, we classify the asymptotic behavior in all possible regimes including heterogeneous natural frequencies and inherent amplitudes, and in particular we identify and characterize a novel regime of \emph{leader-driven synchronization}, wherein one active oscillator can entrain another regardless of frequency mismatch. For general , we prove exponential phase synchronization under sectorial initial data and establish sharp conditions for global amplitude death. Finally, we analyze a real-valued reduction of the model, connecting the dynamics to nonlinear opinion formation and consensus processes. Our results highlight the fundamental differences between amplitude-phase and phase-only Kuramoto models, and provide a new framework for understanding synchronization in heterogeneous oscillator networks.

Paper Structure

This paper contains 19 sections, 26 theorems, 213 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Statement of Results
  3. Two oscillator results
  4. Finitely many oscillator case
  5. The twin oscillators case
  6. Introducing $\alpha$-Heterogeneity
  7. Subcritical Oscillators
  8. Interlude 1: Obtaining the Active/Amplitude Death curve
  9. Mixed oscillators--Stability of the Active Phase-Locked states
  10. Interlude 2: The subcritical on average to supercritical on average transition
  11. Interlude 3: Obtaining the Phase-Locking/Incoherence Curve
  12. Supercritical Oscillators
  13. Homogeneous natural frequency coupled Stuart-Landau Oscillators
  14. Sectorial Solutions
  15. Mixed parameters
  16. ...and 4 more sections

Key Result

Theorem 2.6

Let $N=2$, $\alpha_1=\alpha_2=\alpha$, with $\omega_1=-\omega_2=\omega\geq 0$. Then for any parameter configuration $\alpha \in \mathbb{R},\kappa>0,\gamma\geq 0$ the following represents an invariant manifold: Furthermore, for any initial data $\{z_j(0)\}_{j=1}^2\in \mathcal{M}$, solutions to eq:N=2,a=0,1-eq:N=2,a=0,2 converge to one of the 4 types of asymptotic states seen in Figures fig:Kur2a<0

Figures (9)

  • Figure 1: Phase diagram for the Stuart-Landau system with $N=2$ oscillators with identical subcritical Hopf parameter $\alpha_1=\alpha_2=\alpha<0$. The line $\kappa=\gamma$ is the onset of synchronization where along this line convergence to phase-locking occurs at an algebraic rate, above the line (Tessellated pattern) phase-locking occurs at an exponential rate, and below the line (Solid) the system tends towards a periodic orbit. As $\alpha\leq0$ both oscillators tend to Amplitude Death (Red) at an exponential rate (algebraic at $\alpha=0$). The letters A and B signify the dynamics for the particular choice of parameters in the phase diagram. The first row shows the amplitude death behavior as $|z|\to 0$. The second row shows the phases of each oscillator $\phi_j$ in the first case where Phase-Locking occurs, and the second giving periodic motion.
  • Figure 2: Phase diagram for the Stuart-Landau system with $N=2$ oscillators with identical supercritical Hopf parameter $\alpha_1=\alpha_2=\alpha>0$. The line $\kappa=\gamma$ is the onset of synchronization where along this line convergence to phase-locking occurs at an algebraic rate, above the line (Tessellated) phase-locking occurs at an exponential rate, and below the line (Solid) the system tends towards a periodic orbit. The curve $\kappa^*(\gamma)$ as defined in \ref{['eq:AAcurve1']}, and the line $\kappa=2\alpha$ for $\gamma>2\alpha$, determine whether or not oscillators remain Active (Green) or tend to Amplitude Death (Red). The letters A,B,C,D signify a particular choice of parameters in the phase diagram for which the dynamics are shown.
  • Figure 3: Phase diagram for the Stuart-Landau system with $N=2$ oscillators with nonidentical subcritical Hopf parameters $\alpha_2<\alpha_1\leq0$. The only asymptotic state is Phase-Locked--Amplitude Death (Tessellated, Red). The dynamics for the two points A and B are given where both lead to Amplitude Death and Phase-Locking, but the increased rotation speed in B can be seen.
  • Figure 4: Phase diagram for the Stuart-Landau system with $N=2$ oscillators with nonidentical, on average subcritical, Hopf parameters $-\alpha_2>\alpha_1>0$. The phase behavior remains Phase-Locked (Tessellated), while the Active state (Green) is recovered for low coupling values such that $(\kappa,\gamma)$ is below the curve $f(\alpha_1,\alpha_2,\kappa,\gamma)=0$, and Amplitude Death (Red) above the curve. The three points A,B,C give dynamics for those particular parameter configurations. Notably one can see the increased rotation speed of the Phase-Locked states when comparing C (small $\kappa$, large $\gamma$) to A (small $\kappa$, small $\gamma$). Point B highlights the ability to track the phase-locking behavior through amplitude death.
  • Figure 5: Phase diagram for the Stuart-Landau system with $N=2$ oscillators with nonidentical, on average critical, Hopf parameters $\alpha_1=-\alpha_2>0$. The phase behavior remains Phase-Locked (Tessellated), while the Active state (Green) is recovered for $(\kappa,\gamma)$ below the curve $f(\alpha_1,\alpha_2,\kappa,\gamma)=0$, and Amplitude Death (Red) above the curve. The points A,B,C give dynamics for the chosen parameters in the phase diagram. Note the qualitative differences: (A-small $\kappa$, small $\gamma$) slow rotations with medium difference in amplitudes, (B-small $\kappa$, large $\gamma$) fast rotations, large difference in amplitudes, (C-large $\kappa$, large $\gamma$) fast amplitude death.
  • ...and 4 more figures

Theorems & Definitions (48)

  • Definition 2.1: Amplitude and Frequency Heterogeneity
  • Definition 2.2: Phase Difference and Average
  • Definition 2.3: Amplitude Ratios
  • Definition 2.4: Active State versus Amplitude Death
  • Definition 2.5: Phase-Locking versus Incoherence (Periodic Orbit)
  • Theorem 2.6
  • Theorem 2.7
  • Definition 2.8: Disagreement, Compromise, and Consensus
  • Theorem 2.9
  • Theorem 3.1
  • ...and 38 more