On the Critical Coupling for Kuramoto Oscillators

Abstract: The Kuramoto model captures various synchronization phenomena in biological and man-made systems of coupled oscillators. It is well-known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength for finite and infinite-dimensional, and for first and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. Fourth and finally, we extend our analysis to multi-rate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that the multi-rate Kuramoto model is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies, and we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization. Interestingly, these conditions do not depend on the inertiae which contradicts prior observations on the role of inertiae in synchronization of second-order Kuramoto models.

• The paper establishes explicit necessary and sufficient conditions for synchronization by introducing phase cohesiveness as a key metric.
• It conducts a comprehensive review and derives the threshold Kcritical = ωmax − ωmin for finite-dimensional Kuramoto models.
• The research extends its analysis to multi-rate models, showing that synchronization conditions remain invariant to inertial effects.

An Analysis on Critical Coupling in Kuramoto Oscillators

The study of synchronization phenomena is fundamental in understanding the dynamics of coupled oscillators, impacting fields from biological rhythms to technological systems. This paper explores the Kuramoto model, a well-regarded framework for analyzing synchronization in systems of coupled oscillators, and focuses on determining the critical coupling strength necessary for achieving synchronization.

Main Contributions

The authors present four significant contributions in their exploration of critical coupling for Kuramoto oscillators:

1. Conceptual Framework for Synchronization: The paper begins by distinguishing various synchronization concepts traditionally used in the literature. It introduces the notion of phase cohesiveness as a metric and analytical tool for understanding synchronization, complementing existing frameworks such as phase and frequency synchronization.
2. Comprehensive Review of Critical Coupling Literature: A thorough literature review on estimates of critical coupling strength is provided, addressing both finite and infinite-dimensional models along with first and second-order differential equations. This review sets the stage for the novel contribution of the first explicit condition for necessary and sufficient critical coupling in finite-dimensional Kuramoto models with arbitrarily distributed natural frequencies.
3. Establishment of a New Sufficient and Necessary Condition: The cornerstone of the paper is the derivation of explicit conditions for synchronization under finite-dimensional settings. This involves characterizing a critical coupling strength $K_{\text{critical}}$, defined as $\omega_{\max} - \omega_{\min}$, where oscillators achieve synchronization provided $K > K_{\text{critical}}$. This result is significant for providing practical stability insights, delineating initial conditions for phase cohesiveness, and projecting long-term behavior of the order parameter.
4. Extension to Multi-Rate Kuramoto Models: The analysis is extended to systems involving mixed first-order and second-order dynamics (multi-rate models). The authors prove that these heterogeneous networks can be effectively related back to a first-order Kuramoto model with scaled natural frequencies. A noteworthy finding is that synchronization conditions are invariant to inertia, contradicting some prior studies that suggested inertial effects play a significant role in synchronization dynamics.

Implications and Future Directions

The implications of this research are both practical and theoretical. Practically, the explicit synchronization conditions offer a usable criterion for engineers and scientists working with systems characterized by coupled oscillatory behavior. The conditions provide a bounded region within which synchronization is guaranteed, aiding in effective system design and stability analysis. Theoretically, this work invites further examination into other complex network dynamics where extension of these principles could unveil additional deterministic factors affecting synchronization.

Future studies can expand on this work by exploring non-uniform and time-varying couplings, phase lags, or systems where chaotic dynamics are present. Additionally, the alignment with other synchronization models such as those pertaining to the grid or neural networks opens potential interdisciplinary applications.

Conclusion

This paper makes a substantial contribution to the understanding of synchronization in Kuramoto oscillators, providing new insights into the necessary and sufficient conditions for synchronization across various models. It extends theoretical understanding and has significant implications for both existing and future technological systems employing coupled oscillator dynamics. The findings challenge previous notions regarding the role of inertial effects, adding to the discourse on how similar systems may be approached in future research.