Synchronization in Networks of Heterogeneous Kuramoto-Sakaguchi Oscillators with Higher-order Interactions

Abstract: How do the combined effects of phase frustration, noise, and higher-order interactions govern synchronization in globally coupled heterogeneous Kuramoto oscillators? To address this question, we investigate a globally coupled network of Kuramoto-Sakaguchi oscillators that includes both pairwise (1-simplex) and higher-order (2-simplex) interactions, together with additive stochastic forcing. Systematic numerical simulations across a broad range of coupling strengths, phase-lag values, and noise intensities reveal that synchronization emerges through a nontrivial interplay among these parameters. In general, weak frustration combined with mutually reinforcing coupling promotes synchronization, whereas strong frustration favors coherence under repulsive coupling. Forward and backward parameter sweeps reveal the coexistence of synchronized and desynchronized states. The presence and width of this bistable region depend sensitively on phase frustration, noise intensity, and higher-order coupling strength, with higher-order interactions significantly widening the bistable interval. To explain these behaviors, we employ the Ott-Antonsen reduction to derive a low-dimensional amplitude equation that predicts the forward critical point in the thermodynamic limit, the backward saddle-node point, and the width of the bistable region. Higher order interactions widen this region by shifting the saddle-node point without affecting the forward critical point. Further analysis of Kramer's escape rate explains how noise destabilizes coexistence states and diminishes bistability. Overall, our results provide a unified theoretical and numerical framework for frustrated, noisy, higher-order oscillator networks, revealing that synchronization is strongly influenced by the combined action of phase frustration, stochasticity, and both pairwise and higher-order interactions.

• The paper demonstrates that higher-order interactions robustly enlarge bistable synchronization regions in SK oscillator networks.
• It employs a combined numerical approach and an Ott–Antonsen reduction to show how noise and phase frustration shift transition thresholds.
• The findings suggest that multi-body couplings can counteract desynchronizing effects, offering insights for neural circuits and power grids.

Synchronization Phenomena in Heterogeneous Kuramoto–Sakaguchi Oscillator Networks with Higher-order Interactions

Introduction

This work investigates the emergence and modulation of synchronization in globally coupled networks of heterogeneous Kuramoto–Sakaguchi (SK) oscillators subject to both pairwise (1-simplex) and higher-order (2-simplex) interactions, with additive noise and phase-lag-induced frustration. Prior studies have analyzed the impacts of frustration or noise or higher-order interactions separately; however, their joint effect, especially in the presence of nontrivial multi-body couplings, remains poorly characterized. The authors implement systematic numerical and analytical studies to elucidate the interplay among coupling topologies, stochastic perturbations, and intrinsic phase frustration in shaping synchronization and multistability.

The significance of this investigation is established by the pervasiveness of synchronization phenomena in physics, neuroscience, power grid engineering, and complex social systems. The Kuramoto framework, extended here through the Sakaguchi phase lag and non-pairwise interactions, enables modeling of real-world networks with simultaneous multi-oscillator couplings. This generalization is formalized via simplicial complexes (Figure 1).

Figure 1: Architectural depiction of globally coupled SK oscillator network, indicating 0-simplex nodes, pairwise edges (1-simplices), and higher-order triangles (2-simplices) as coupling motifs.

Model, Parameter Space, and Numerical Exploration

The governing equations (Eq. 1) include conventional sinusoidal pairwise interaction (strength $k_1$), 2-simplex higher-order coupling (strength $k_2$), additive Gaussian white noise (intensity $D$), and the Sakaguchi phase frustration parameter $\beta$. Oscillator frequencies are heterogeneous, sampled from a Lorentzian distribution.

Order parameter dynamics are analyzed with respect to variations in $k_1$, $k_2$, $\beta$, and $D$ across the network. The first set of numerical experiments sweeps $(k_1, \beta)$ for selected $k_2$ and $D$. Figure 2 presents the resulting synchronization landscapes, mapping the degree of macroscopic coherence $R$.

Figure 2: Synchronization landscapes illustrating regimes of coherence ($R \approx 1$) and incoherence ($R \approx 0$) as functions of pairwise coupling $k_1$, frustration $\beta$, higher-order strength $k_2$, and noise $D$.

Key findings include:

• For weak frustration ($\beta \in (-\pi/2, \pi/2)$), positive $k_1$ and $k_2$ promote synchronization.
• Strong frustration or negative $\beta$ require negative (repulsive) $k_1$, or amplified $k_2$, to stabilize coherent dynamics.
• Stochasticity ($D > 0$) suppresses and smooths synchronization transitions, transforming abrupt (first-order) jumps into continuous (second-order) bifurcations.

Subsequent analyses elucidate hysteresis and multistability using forward- and backward-sweep protocols in $k_1$, shown in Figures 3 and 4 for representative $k_2$ and $\beta$.

Figure 3: Hysteretic response of the order parameter $R$ under forward (magenta) and backward (cyan) sweeps of $k_1$, highlighting regions of bistability and the effect of noise and frustration.

Figure 4: Similar to Figure 3 but with elevated higher-order coupling $k_2=20$, showing pronounced widening of the bistable region.

Distinct bistable intervals (with coexisting synchronized and desynchronized attractors) emerge, whose width is maximized for strong higher-order coupling and minimal noise/frustration.

Analytical Reduction: Ott–Antonsen Ansatz and Bifurcation Structure

The authors employ the Ott–Antonsen ansatz to obtain a tractable low-dimensional reduction in the thermodynamic limit. The amplitude equation (Eq. 12) for the order parameter $r = |z_1|$ incorporates the combined effects of pairwise and 2-simplex terms, frustration, and stochasticity.

Comparison of the reduced theory with full numerical simulations is shown in Figure 5, verifying accuracy across parameters.

Figure 5: Agreement between theoretical and simulated order parameter amplitudes $R$ as a function of $k_1$ for multiple $D$ and $\beta$; vertical jumps indicate first-order transitions.

Fixed-point and linear stability analysis reveals:

• The trivial incoherent state ($r^* = 0$) is always stable when $k_1$ is below a critical threshold $k_{1c}$.
• Nontrivial synchronized states bifurcate through either subcritical or supercritical pitchfork transitions, depending on parameters.
• The forward transition corresponds to the loss of stability of the incoherent state (at $k_{1c}$), while the backward transition is governed by a saddle-node bifurcation (at $k_1^{\text{SN}}$).

The width of the bistable region, $\Delta k = |k_{1c}-k_1^{\text{SN}}|$, is found to increase monotonically with $k_2$ and decrease with $D$ and $\beta$. Figure 6 quantitatively illustrates this dependence.

Figure 6: Hysteresis width $\Delta k$ as a function of higher-order coupling $k_2$ for multiple $D$ and $\beta$; higher-order interactions robustly expand multistability.

Kramers' escape theory is applied to rationalize the role of noise in facilitating transitions between metastable basins, thereby narrowing multistability domains.

Theoretical and Practical Implications

This research demonstrates that higher-order coupling is a primary control knob for both the existence and breadth of synchronization and multistability in oscillator systems—with increasing $k_2$ not only shifting the onset of coherence to lower $k_1$, but also expanding the bistable regime. These findings directly contradict interpretations that finite-size or pairwise topological randomness alone can regulate the extent of multistability. Instead, structure and strength of non-pairwise interactions are decisive.

The inclusion of noise and frustration reflects practical scenarios (biological populations, neural circuits, power grids) where intrinsic stochasticity and phase-lag coexist with multi-body effects. The insight that higher-order coupling can counteract the desynchronizing influence of both stochastic fluctuations and intrinsic frustration is immediately pertinent for the design of engineered synchrony in distributed hardware, and for understanding robustness of collective phases in natural systems.

From a theoretical perspective, the approach unifies prior developments in higher-order network theory and stochastic synchronization into a common analytic framework. The Ott–Antonsen reduction remains valid even in the presence of arbitrary higher-order deterministic coupling and additive stochasticity, allowing extension to more complex network architectures, time-delays, or adaptive simplicial couplings.

Conclusion

The study rigorously elucidates the role of higher-order (2-simplex) interactions, phase frustration, and stochasticity on synchronization landscapes and multistability in SK oscillator networks. Key numerical and analytic results demonstrate that higher-order coupling robustly enlarges the bistable (coexistence) region and shifts synchronization thresholds, while additive noise and frustration suppress and smooth coherence. Analytical reduction precisely predicts critical points, hysteresis, and the modulation of metastable lifetimes via Kramers' escape rate. These results have broad implications for networks where non-pairwise interactions and noise collectively determine the emergence and resilience of macroscopic synchrony. The analytic and computational framework provided will facilitate future extensions to spatially distributed, heterogeneous, or adaptive oscillator systems.

[See: "Synchronization in Networks of Heterogeneous Kuramoto-Sakaguchi Oscillators with Higher-order Interactions" (2512.10593)]