Frequency Heterogeneity can Promote Order yet Undermine Stability in Kuramoto Networks with Higher-Order Interactions

Abstract

We investigate the interplay between frequency heterogeneity and higher-order triadic interactions in a ring network of Kuramoto oscillators. While both factors individually disrupt ordered states, their combination produces unexpected collective behavior. In the strong triadic coupling regime, moderate frequency heterogeneity substantially increases the global order parameter, with an optimal heterogeneity strength growing approximately linearly with triadic coupling strength. Basin stability analysis reveals that this order-promoting effect arises from a global restructuring of the attractor landscape: frequency heterogeneity shifts the attractor competition in favor of more ordered configurations. Linear stability analysis of frequency-locked twisted states reveals a competing effect: frequency heterogeneity monotonically erodes linear stability and reduces the probability of frequency locking. These two competing mechanisms, basin enlargement and linear destabilization, together account for the non-monotonic dependence of the order parameter on heterogeneity strength. Our results demonstrate that frequency heterogeneity can play a constructive role in oscillator networks with higher-order interactions.

Figures (5)

• Figure 1: Schematic of the ring network ($n=12$, $r=2$). (a) Global topology with pairwise interactions: green lines highlight the couplings between oscillator $i$ (red) and its four nearest neighbors (green nodes); black lines show remaining network edges. (b) Triadic interactions acting on oscillator $i$: each shaded triangle shares $i$ as a vertex, with $j$ and $k$ being two distinct neighbors; there are $\binom{2r}{2}=6$ such triangles in total. Green solid lines indicate pairwise coupling; orange dashed lines and shading indicate triadic coupling.
• Figure 2: Effect of frequency heterogeneity on the global order parameter $R$. (a) Heat map of $R$ in the $(\sigma_{\Delta}, \omega_{std})$ parameter space. The red curve marks the optimal heterogeneity $\omega_{std}^{*}$ that maximizes $R$ at each $\sigma_{\Delta}$. Insets show representative phase profiles of the three collective states: twisted state (top), chimera state (middle), and disordered state (bottom). (b) Cross-sectional curves of $R$ versus $\sigma_{\Delta}$ at four values of $\omega_{std}$. Bar chart shows the standard deviation of $R$ across independent realizations (right axis). Parameters: $n = 83$, $r = 2$, $\sigma = 1.0$, with $16{,}000$ independent realizations.
• Figure 3: Ordered fraction ($R > 0.6$) as a function of frequency heterogeneity $\omega_{std}$, measured from low-order initial conditions ($R < 0.6$). (a) Fixed frequency distribution shape, $1{,}600$ different initial conditions. (b) Fixed initial condition, $1{,}600$ independent frequency realizations. Parameters: $n = 83$, $r = 2$, $\sigma = 1.0$, $\sigma_{\Delta} = 5$.
• Figure 4: Change in order parameter $\Delta R = R(\omega_{std}) - R_0$ as a function of $\omega_{std}$, where $R_0 = R(0)$ is the order parameter of the same initial condition in the homogeneous system. Box plots show the interquartile range (IQR, 25%--75%), median (red line), and mean (red point) across $1{,}600$ independent random samples at each $\omega_{std}$. Error bars indicate the standard deviation. The dashed line marks $\Delta R = 0$. Parameters: $n = 83$, $r = 2$, $\sigma = 1.0$, $\sigma_{\Delta} = 5$.
• Figure 5: Linear stability of frequency-locked twisted states for $q = 0, 1, 2$. Upper panels (a)--(c): fraction of locked states as a function of $\omega_{std}$. Lower panels (d)--(f): MTLE $\lambda_{\max}$ for each of the $160$ frequency realizations; each curve corresponds to one frequency shape tracked across $\omega_{std}$. The dashed line marks $\lambda_{\max} = 0$. Parameters: $n = 83$, $r = 2$, $\sigma = 1.0$, $\sigma_{\Delta} = 5$.