Cluster dynamics in a two-group Stuart-Landau model analyzed by the second-order phase reduction

Abstract: We analyze cluster states in an ensemble of Stuart-Landau oscillators with two subpopulations of different frequencies. Our main goal is to compare the descriptions of the system's dynamics obtained via the standard first-order phase approximation and the second-order phase reduction. We demonstrate that the second-order model not only provides quantitative improvements in the description but also reveals new dynamical states not present in the standard Kuramoto theory. In particular, it describes bistability of synchronous states in the minimal setup of two coupled oscillators and the existence of three-cluster states, forbidden in the first-order phase description by the Watanabe-Strogatz theory. The very good agreement between the second-order approximation results and the results of numerical simulations of the original Stuart-Landau network highlights the usefulness of high-order phase-reduction models.

• The paper demonstrates that second-order phase reduction predicts phenomena absent from the first-order Kuramoto model, including in-phase/anti-phase bistability near α = π/2 and stable three-cluster states.
• The analysis combines a second-order Adler equation, transversal Lyapunov exponents, and bifurcation analysis to map locking and cluster stability, with predictions closely matching simulations of 100-oscillator Stuart–Landau networks.
• The results show that quadratic coupling terms become essential when coupling is strong or limit cycles are weakly stable, while phase-reduction validity and large-cluster assumptions limit the predicted stability regions.

Overview

This paper by Baibolatov, Omel'chenko, and Rosenblum examines cluster states in a globally coupled ensemble of isochronous Stuart–Landau (SL) oscillators divided into two equal-size subpopulations with distinct natural frequencies. The central methodological contribution is a systematic comparison between the standard first-order phase reduction—which yields the Kuramoto model—and the second-order phase reduction derived from the framework of Mau, Omel'chenko, and Rosenblum (2606.04668). The authors demonstrate that the second-order model is not merely a quantitative refinement: it predicts qualitatively new phenomena absent from the first-order description, including bistability of in-phase and anti-phase locking in the minimal two-oscillator setup and stable three-cluster states that are forbidden by the Watanabe–Strogatz theory within the first-order Kuramoto framework.

Model and second-order phase reduction

The system consists of $N = 2N_g$ SL oscillators with global diffusive coupling characterized by strength $\varepsilon$ and phase lag $\alpha$, with frequencies drawn from a bimodal delta distribution ($\omega_1 > \omega_2$). Applying the second-order phase reduction of (2606.04668) yields phase equations containing terms of order $\varepsilon$ and $\varepsilon^2/\kappa$, where $\kappa = -2\eta < 0$ is the Floquet exponent quantifying the stability of the individual limit cycle. The second-order terms include triplet interaction terms and pairwise coupling terms for non-connected units—features with no counterpart in the Kuramoto model. Under the assumption $|\omega^{(n)} - \omega^{(m)}| \ll |\kappa|$, the authors adopt the simplified form due to León and Pazó.

The relevant control parameter is the ratio $e = \varepsilon/|\kappa|$; second-order effects become essential for strong coupling and/or weakly stable limit cycles. The most consequential regime is $\alpha \approx \pi/2$, where the first-order coupling is nearly neutral and the quadratic terms dominate the dynamics.

Two coupled oscillators: bistability from the second-order Adler equation

For $\varepsilon$0, the phase difference $\varepsilon$1 obeys a second-order generalization of the Adler equation:

$\varepsilon$2

where $\varepsilon$3. The analysis is fully analytical. At $\varepsilon$4 the first-order term vanishes entirely—no synchrony exists at any $\varepsilon$5 in the first approximation—yet the second-order equation yields locking with condition $\varepsilon$6 and, crucially, bistability: if $\varepsilon$7 is a locked solution, so is $\varepsilon$8, so in-phase states ($\varepsilon$9) and anti-phase states coexist.

For general $\alpha$0, solving the extremum condition reduces to a quadratic in $\alpha$1 with roots $\alpha$2, where $\alpha$3. Both roots can satisfy $\alpha$4 simultaneously, producing an overlap interval $\alpha$5 (with $\alpha$6) in which in-phase and anti-phase Arnold tongues coexist. The anti-phase tongue requires a threshold coupling $\alpha$7, which vanishes at $\alpha$8 but grows rapidly away from it—for $\alpha$9 it reaches roughly $\omega_1 > \omega_2$0, already outside the validity range of the phase approximation. The authors state plainly that bistability is therefore practically confined to a narrow interval around $\omega_1 > \omega_2$1. A further qualitative departure from first-order theory is that the span of the phase shift across each tongue, $\omega_1 > \omega_2$2, becomes dependent on both $\omega_1 > \omega_2$3 and $\omega_1 > \omega_2$4, rather than fixed at $\omega_1 > \omega_2$5. Numerical simulations of the original SL system confirm the analytical tongue boundaries.

Two-cluster states and transversal stability

For $\omega_1 > \omega_2$6, the $\omega_1 > \omega_2$7 state—each group forming one cluster—reduces to the same second-order Adler equation for the inter-cluster phase difference, but with an additional constraint: clusters must be transversally stable against evaporation of individual oscillators. Using virtual test oscillators that experience the mean field without contributing to it, the authors derive instantaneous growth rates $\omega_1 > \omega_2$8 whose time averages give the transversal Lyapunov exponents (TLEs).

In the first-order approximation, the analysis is fully explicit. Synchronous anti-phase cluster states are shown to be unconditionally transversally unstable. Synchronous in-phase states are stable throughout their locking domain for $\omega_1 > \omega_2$9, but lose stability at critical detuning $\varepsilon$0 for $\varepsilon$1. Asynchronous $\varepsilon$2 states are stable for $\varepsilon$3 and for $\varepsilon$4 otherwise.

In the second-order approximation, stability boundaries are obtained semi-analytically by numerically solving $\varepsilon$5 together with the fixed-point condition, and $\varepsilon$6 for drifting solutions. The resulting stability diagram agrees closely with dynamical continuation performed directly on the full SL system with $\varepsilon$7. Notably, the second-order model corrects the first-order prediction near $\varepsilon$8: since the first-order tongues collapse there while the true SL system exhibits locking, only the quadratic terms capture the observed behavior.

Three-cluster states forbidden in first-order theory

The paper's most striking result concerns the $\varepsilon$9 state, in which the faster group splits into two equally sized clusters while the slower group remains coherent. Such configurations cannot exist in the first-order Kuramoto model: the Watanabe–Strogatz theory implies that identical globally coupled phase oscillators cannot split into more than two clusters per population. Yet they occur in the SL network, and only the second-order reduction explains them.

A linear change of variables shows that the three-cluster dynamics is effectively two-dimensional on a torus parameterized by the intra-group and inter-group phase differences. Varying $\varepsilon^2/\kappa$0 reveals a clear bifurcation scenario: for small $\varepsilon^2/\kappa$1, only the symmetric $\varepsilon^2/\kappa$2 solution is stable; as $\varepsilon^2/\kappa$3 increases, the unstable asymmetric branch undergoes a subcritical pitchfork bifurcation of limit cycles, giving birth to one stable asymmetric ($\varepsilon^2/\kappa$4) solution flanked by two unstable ones. These unstable branches act as separatrix boundaries, enabling bistability between the $\varepsilon^2/\kappa$5 and $\varepsilon^2/\kappa$6 states. The bifurcation threshold was traced numerically via continuation on the order parameter $\varepsilon^2/\kappa$7, which jumps abruptly to unity at the critical point.

Transversal stability of the $\varepsilon^2/\kappa$8 state was assessed via the TLEs of all three clusters; typically the slowest cluster loses stability first. The combined macroscopic and transversal stability region divides parameter space into three zones: only $\varepsilon^2/\kappa$9 states, bistability of both, or only $\kappa = -2\eta < 0$0 states. Direct simulation of the full SL system with $\kappa = -2\eta < 0$1 reproduces these boundaries accurately—an important validation given that the analysis rests on the assumption that clusters are large enough that single-oscillator evaporation negligibly perturbs the mean field.

Limitations and open questions

Several caveats qualify the results. The phase reduction itself is valid only for sufficiently small $\kappa = -2\eta < 0$2; some predicted thresholds (e.g., $\kappa = -2\eta < 0$3 at $\kappa = -2\eta < 0$4) fall outside this range, restricting observable bistability to a narrow window around $\kappa = -2\eta < 0$5. The cluster stability analysis assumes large clusters ($\kappa = -2\eta < 0$6), neglecting order-$\kappa = -2\eta < 0$7 mean-field fluctuations. All observed $\kappa = -2\eta < 0$8 states had equally sized clusters within each group, but the analysis does not establish whether unequal partitions or higher $\kappa = -2\eta < 0$9 combinations exist—the authors report not having observed them but do not rule them out. Finally, the authors note that third- and higher-order reductions are constructible in principle but resist analytical treatment and offer no computational advantage over simulating the original system, leaving open whether intermediate orders provide useful compromises.

Conclusion

This work provides concrete evidence that second-order phase reduction yields genuine predictive power beyond quantitative corrections to Kuramoto theory. For a two-frequency SL population, the quadratic terms analytically explain in-/anti-phase bistability near neutral coupling and account for three-cluster states prohibited by Watanabe–Strogatz integrability, with stability boundaries matching direct simulations of the full oscillator network. The study positions high-order phase models as practical analytical tools for oscillatory networks rather than formal extensions, while delineating precisely where their validity ends.