Second-order Kuramoto model with adaptive simplicial complex

Abstract: We investigate the emergence of synchronization in the second-order Kuramoto model with adaptive simplicial interactions on a globally connected network. This inertial Kuramoto framework describes systems, where oscillator frequencies evolve over time. Unlike most previous work that ignores inertia, we examine how inertia combined with adaptive higher-order coupling alters synchronization transitions. Using self-consistency analysis, we derive the steady-state behavior and show that adaptation qualitatively reshapes the synchronization landscape. We find that the backward transition from synchronization to incoherence remains controlled by the adaptive feedback parameter, but the forward discontinuous jump to synchronization vanishes in the thermodynamic limit. In contrast, finite-size systems still display an abrupt transition to synchronization, with its onset precisely set by the adaptation control parameter. These results show how adaptive feedback and system size together govern the onset and robustness of synchronization in inertial oscillator networks with higher-order interactions.

• The paper introduces an adaptive feedback mechanism for both pairwise and triadic couplings in a second-order Kuramoto framework.
• It analytically and numerically demonstrates that adaptation suppresses abrupt transitions in large oscillator networks.
• The model reveals how system size, inertia, and noise influence multistability and synchronization criticalities in real-world systems.

Adaptive Simplicial Interactions in the Second-Order Kuramoto Model: Synchronization Dynamics and Critical Transitions

Introduction

This work addresses synchronization phenomena in oscillator networks by extending the second-order Kuramoto model to encapsulate adaptive couplings and higher-order (simplicial) interactions. Inertia, commonly omitted in standard Kuramoto formulations, is incorporated to more accurately capture the dynamical behavior of real-world systems such as power grids and neuronal populations. The primary novelty here lies in the adaptive feedback mechanism, modulating both pairwise and triadic couplings based on collective synchronization, and the rigorous analysis of these effects on synchronization transitions—both analytically in the continuum limit and numerically in finite-size systems.

Model Framework and Analytical Formulation

The analyzed system consists of $N$ globally coupled phase oscillators, each described dynamically by

$$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$

where $m$ is the inertia, $K_1$ and $K_2$ are strengths for pairwise and triadic (2-simplex) interactions respectively, and $\gamma$ is the adaptation exponent modulating the feedback strength according to the global synchronization measure $r_1$. The adaptation mechanism ensures that the effective network couplings become dynamically contingent on the instantaneous degree of synchronization, formally $K_{1,2}^{\text{eff}} = K_{1,2} r_1^{\gamma}$. The global order parameter $r_1$ quantifies the level of synchrony, while $r_2$ tracks clustering effects associated with multi-modal distributions in phase space. Intrinsic frequencies $$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$0 follow a Lorentzian distribution.

The mean-field reduction facilitates self-consistency analysis, yielding analytical predictions for the existence and stability of incoherent, weakly synchronized, and fully synchronized states in the thermodynamic limit. The reduction leads to a single-oscillator equation analogous to the Josephson junction, which is scrutinized for fixed points, limit cycles, and associated bifurcation scenarios.

Figure 1: Schematic diagram of the second-order Kuramoto model with adaptive pairwise and triadic couplings, displaying locked and drifting oscillators and illustrating the adaptation controlled by $$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$1 and exponent $$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$2.

Synchronization Transitions: Thermodynamic Limit and Finite Size Effects

Key analytical results show that with adaptation ($$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$3), the incoherent state remains stable for all coupling strengths in the continuum limit ($$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$4). This precludes the abrupt first-order transition to synchrony typically observed in the non-adaptive ($$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$5) second-order Kuramoto model. However, for finite-size systems, stochastic fluctuations in $$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$6 provide a mechanism for the system to escape the basin of attraction of the incoherent state, resulting in an abrupt, fluctuation-driven transition to a weakly synchronized state.

Figure 2: Phase diagram in the $$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$7 plane illustrating regions of fixed point and limit cycle solutions, and (b) $$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$8 versus $$m\ddot{\theta}_i = -\dot{\theta}_i + \Omega_i + K_1 r_1^{\gamma}\sum_k \sin(\theta_k - \theta_i) + K_2 r_1^{\gamma} \sum_{k,l} \sin(2\theta_k - \theta_l - \theta_i)$$9 highlighting the absence or presence of a forward jump depending on $m$0 and system size.

As $m$1 increases, the amplitude of fluctuations in $m$2 scales down as $m$3, leading the threshold for the forward transition to higher values of $m$4 and effectively suppressing the abrupt jump in the thermodynamic limit.

Figure 3: Dependence of synchronization transitions on system size $m$5, inertia $m$6, and the onset of multistability—demonstrating the forward jump vanishing with increasing $m$7 and varied $m$8 affecting $m$9 plateau heights.

Multistability and Hysteresis

The model exhibits pronounced multistability due to the coexistence of incoherent and synchronized attractors, especially for large inertia and intermediate coupling. Forward and backward numerical continuations reveal distinct transition points ($K_1$0 and $K_1$1, respectively), with their separation quantifying the width of the bistable, hysteretic region.

Figure 4: Shifting of $K_1$2 as functions of $K_1$3 and $K_1$4—adaptation exponent $K_1$5 is observed to shift both forward and backward transitions, whereas $K_1$6 mainly impacts the backward transition.

Analytical predictions for the position of these branches, derived via self-consistency under appropriate frequency limits, agree well with simulation results. The multistable regime appears as a set of branches located between analytically determined bounding curves.

Parametric Dependence of Transition Points

A systematic exploration demonstrates:

• System size $K_1$7 and inertia $K_1$8: Only the forward transition point $K_1$9 increases with $K_2$0 and $K_2$1.
• Triadic coupling $K_2$2: Only the backward transition point $K_2$3 is lowered by increasing $K_2$4.
• Adaptation exponent $K_2$5: Both $K_2$6 and $K_2$7 increase with $K_2$8, reflecting a weakening of effective coupling for larger adaptation exponents.

Figure 5: Dependence of $K_2$9 and $\gamma$0 on $\gamma$1, $\gamma$2, $\gamma$3, and $\gamma$4—demonstrating nuanced roles of system parameters and adaptation in shaping critical transitions.

This separation of influence underscores how adaptation can selectively modulate the network's vulnerability and resilience to synchronization as a function of dynamical and structural factors.

Fluctuations, Noise, and Robustness

Introducing additive Gaussian white noise further modulates transition dynamics. Increased noise strength $\gamma$5 consistently shifts both forward and backward transitions to higher $\gamma$6 values, making synchronization less accessible. For sufficiently large $\gamma$7, the distinction between weakly synchronized and fully synchronized branches becomes blurred.

Figure 6: $\gamma$8 versus $\gamma$9 with varying noise strengths $r_1$0, illustrating noise-induced shifts of synchronization transitions and eventual merging of branches at higher $r_1$1.

Implications and Future Directions

The findings have direct theoretical and applied implications:

• Power grids and neuronal networks: The framework models adaptive restructuring of coupling during synchronization, relevant to load-dependent transmission in power networks and activity-driven plasticity in neural circuits.
• Robust adaptation: The adaptive scheme resists abrupt global synchronization under parameter drift, which in engineered systems could mitigate risks of sudden cascades or failures.
• Model extension: The analysis, being restricted to globally coupled networks and specific functional forms of adaptation, invites future work on heterogeneous/adaptive topologies and nonlinear feedback.
• Generalizability: The self-consistency approach can be adapted for broader classes of adaptive oscillator models, including those with more general forms of higher-order couplings or adaptation rules.

Conclusion

This study systematically extends the second-order Kuramoto model to include adaptive, simplicial interactions, offering detailed analytic and computational evidence for the profound reshaping of synchronization transitions by adaptation. It reveals that adaptive feedback, via the exponent $r_1$2, couples with system size, inertia, and higher-order interactions to determine the onset and robustness of synchrony, the nature and extent of multistability, and the susceptibility to noise. This work thereby establishes an analytic framework for adaptive synchronization in complex oscillator networks, providing guidance for the design and control of synchronization in real and engineered systems.

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Reference:

"Second-order Kuramoto model with adaptive simplicial complex" (2604.10247)