Inferring Coupled Stuart-Landau Equations from Waveforms

TL;DR

The paper presents a data-driven framework to infer phase–amplitude dynamics of coupled oscillators near a supercritical Hopf bifurcation by reconstructing a near-identity transformation to a Stuart–Landau form. It first uses two observables from an isolated oscillator to recover the nonlinear variable transform and SL parameters $(a,b,c,d)$, then estimates linear coupling terms from paired data, yielding a compact, mechanistic model. Validation on coupled van der Pol oscillators and a high-dimensional hydrodynamic system shows that the inferred SL model captures amplitude-mediated synchronization features, including bistability between in-phase and anti-phase states and a Neimark–Sacker bifurcation destabilizing anti-phase synchronization. The approach requires only short time-series and is flexible to use any two observables, enabling quantitative prediction of synchronization transitions from waveform data in diverse physical systems.

Abstract

We present a data-driven framework to infer phase-amplitude equations of coupled limit-cycle oscillators directly from waveform measurements. Exploiting the universality of the Stuart-Landau normal form near a supercritical Hopf bifurcation, we reconstruct a near-identity transformation from two independent observables of an isolated oscillator and infer the intrinsic Stuart-Landau parameters. Using this reconstructed transformation, we then estimate linear coupling coefficients from paired measurements. The method accurately recovers parameters for coupled van der Pol oscillators, providing a quantitative benchmark. Applied to a high-dimensional hydrodynamic system of two coupled collapsible-channel oscillators, the inferred Stuart-Landau model captures bistability between in-phase and anti-phase synchronization and reveals that the anti-phase state is destabilized through a Neimark-Sacker bifurcation. Our approach enables quantitative prediction of synchronization transitions involving amplitude dynamics from experimentally accessible waveform data.

Figures (7)

• Figure 1: Schematic overview of the proposed approach with data from a hydrodynamic system used for demonstration. (a) Snapshot of the hydrodynamic system. The system shows bistability between in-phase and anti-phase synchronization (see details in (e)). (b) Example time series $\phi(t)$ and $\psi(t)$, which can be arbitrarily chosen observables from the system. (c) Result of the optimal variable transformation, which maps the original variables $(\phi,\psi)$ to the rotation-symmetric coordinates $z=r e^{i \theta}$, eliminating the dependence of amplitude and phase evolution on the intrinsic oscillation period. (d) The estimated model, in which the transformed variables in (c) are expected to follow in a good approximation. Colors indicate the time course in the panels (b) and (d). As will be shown in Fig. \ref{['fig:3']}, analysis of the estimated model in (d) reveals that the loss of the anti-phase synchronized state occurs via a Neimark-Sacker bifurcation. (e) Color map that represents the final phase difference $\tau/T$ after transients in the plane spanned by the initial phase difference $\tau^{(0)}/T$ between the oscillators (vertical axis) and the coupling strength $\delta$ (horizontal axis), the system exhibits bistability between in-phase and anti-phase synchronization for small $\delta$, while the anti-phase state disappears near $\delta \simeq 9$.
• Figure 2: Estimation results for the van der Pol (VDP) oscillator system. (a) Typical time series of $x$ and $\dot{x}$ used to infer the parameters of a single oscillator. (b) Typical time series of $x_k$ and $\dot{x}_k$ ($k = 1, 2$) used to infer the parameters of coupled oscillators. (c) Estimated parameters of a single oscillator plotted against different values of $t_{\rm s}$ at $t_{\rm w}=5$. (d) Stability diagram of the estimation in the $t_{\rm s}$–$t_{\rm w}$ plane for the single oscillator. (e) Estimated interaction parameters between oscillators plotted against $t_s$ at $t_{\rm w}=18$. (f) Stability diagram of the interaction parameter estimation in the $t_{\rm s}$–$t_{\rm w}$ plane. In (c) and (e), symbols represent the estimated values, and solid lines indicate the theoretical values. In (d) and (f), regions with lower cost function values indicate higher estimation stability.
• Figure 3: Stability analysis of the anti-phase synchronized periodic solution, based on the Stuart-Landau oscillator system estimated from time series data of a fluid system. Here, we neglect the non-resonant terms $I^\qty(0)$, $I^\qty(2)\bar{z}_k$, and $I^\qty(4)\bar{z}_\ell$ in Eq. \ref{['coupledSL']}. (a) Real part of the Floquet exponents. (b) Imaginary part of the Floquet exponents. (c) Schematic illustration that shows the bifurcation structure. The stable limit cycle collides with an unstable torus and destabilizes at $\delta = \delta_c \simeq 9.0$.
• Figure S1: Numerical results for the coupled van der Pol (VDP) equations. (a) Phase diagram for the anti-phase synchrony, where the yellow and purple circles indicate stable and unstable anti-phase states, respectively. From the analysis using the coupled Stuart-Landau equations with non-resonance terms neglected, the anti-phase synchrony is stable when all the real values of $\lambda_\mathrm{amp}^{(\mathrm{anti})}$ in Eq. \ref{['lam_anti_amp']} are negative. The stability boundary is shown by the black line (Eq. \ref{['full_PF']}) and red solid line (Eq. \ref{['NS']}). In contrast, using the equation with the adiabatic approximation, the stability boundary is given by $\lambda_\mathrm{ad}^\mathrm{anti} = 0$; i.e. Eq. \ref{['full_PF']}, which is represented by the combination of the black solid and dashed lines. (b, c) Normalized phase difference $\tau/T$ in the final state depending on (ii) $\tau^\qty(0)/T$ and $I_x$ at $I_v=0.03$ and (iii) $\tau^\qty(0)/T$ and $I_v$ at $I_x=0.16$. $\tau^\qty(0)/T$ is the initial normalized phase difference between two oscillators and $T$ is the intrinsic period of the single oscillator.
• Figure S2: $t_{\rm s}$- and $t_{\rm w}$-dependences of the inferred parameters in (a,b) the single system and (c,d) coupled system of the oscillatory flow obtained by numerical simulation. $t_{\rm w}=25$ for the panel (a) and $t_{\rm w}=100$ for the panel (c). $F_{\rm cost}^\qty(3)$ is the cost function to evaluate the robustness of parameters.