{ "id": "eprygp6j1w", "slug": "kuramoto-model", "title": "Kuramoto model", "updated": "2026-10-21", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis" }, { "id": "physical.math.analysis.dynamical_systems_and_chaos", "label": "Dynamical systems, chaos, and ergodic theory", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.analysis.dynamical_systems_and_chaos" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.analysis", "label": "Non-geographic · 1946 to 2000: Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.analysis", "label": "Analysis and mathematical models", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.analysis" } ] } ], "excerpt": "The Kuramoto model, introduced by Yoshiki Kuramoto in 1975, is a mathematical model of synchronization in which coupled phase oscillators entrain into collective oscillation above a critical coupling strength.", "snippet": "The Kuramoto model, introduced by Yoshiki Kuramoto in 1975, is a mathematical model of synchronization in which coupled phase oscillators entrain into collective oscillation above a critical coupling strength.", "node": "physical.math.analysis.dynamical_systems_and_chaos", "markdown": "# Kuramoto model\n\nThe Kuramoto model is a mathematical model of synchronization in which a population of phase oscillators, each rotating at its own natural frequency, couples through the sine of phase differences and entrains into collective oscillation above a critical coupling strength. It is a representative model of coupled phase oscillators and is applied across physical, biological, chemical, and social systems.[1](https://doi.org/10.1103/revmodphys.77.137) It describes self-sustained oscillators with heterogeneous intrinsic frequencies and exhibits a phase transition at a critical coupling beyond which collective behavior is achieved.[2](https://ar5iv.labs.arxiv.org/html/1511.07139)\n\n| Key fact | Value |\n|---|---|\n| Governing equation | \\( \\dot{\\theta}_{k} = \\omega_{k} + \\frac{K}{N}\\sum_{j=1}^{N}\\sin(\\theta_{j}-\\theta_{k}) \\)[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) |\n| Order parameter | \\( Z = R \\cdot e^{i\\phi} = \\frac{1}{N}\\sum_{j} e^{i\\theta_{j}} \\); \\( R=|Z| \\) measures synchrony[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) |\n| Critical coupling (smooth even unimodal \\( g(\\omega) \\)) | \\( K_{c} = 2/[\\pi g(0)] \\); the explicit amplitude relation \\( r = \\sqrt{1-K_{c}/K} \\) for \\( K \\ge K_{c} \\) holds for the Lorentzian case[4](https://web.mit.edu/~jadbabai/www/ESE680/Strogatz_Kuramoto.pdf) |\n| Lorentzian frequencies of half-width \\( \\Delta \\) | \\( K_{c} = 2\\Delta \\); \\( R = \\sqrt{1-2\\Delta/K} \\)[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) |\n| With white noise strength \\( D \\) | the critical coupling depends on the Lorentzian width \\( \\gamma \\)[1](https://doi.org/10.1103/revmodphys.77.137) |\n| Uncorrelated networks | \\( \\lambda_{c} = \\lambda_{\\max}/2 \\), the largest adjacency-matrix eigenvalue[5](https://www.nature.com/articles/srep04783) |\n| Exact low-dimensional reduction | \\( \\dot{Z} = (-\\Delta + i\\hat{\\omega}) \\cdot Z + \\frac{K}{2} \\cdot Z \\cdot (1-|Z|^{2}) \\) for Lorentzian \\( g(\\omega) \\)[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) |\n\n## How it works\n\nEach oscillator carries a phase \\( \\theta_{k} \\) and advances at its intrinsic frequency \\( \\omega_{k} \\), drawn randomly to account for individual heterogeneity; the coupling strength \\( K \\in \\mathbb{R}^{+} \\) measures interaction strength and could, up to time rescaling, be absorbed into the frequencies via \\( \\omega_{i} \\mapsto \\omega_{i}/K \\).[6](https://royalsocietypublishing.org/rspa/article/474/2220/20180467/80056/The-mathematics-of-asymptotic-stability-in-the) Coupling is equally weighted, all-to-all, and purely sinusoidal, with the factor \\( 1/N \\) ensuring the model is well behaved as \\( N \\to \\infty \\); the frequencies are drawn from a density \\( g(\\omega) \\) assumed unimodal and symmetric about its mean.[4](https://web.mit.edu/~jadbabai/www/ESE680/Strogatz_Kuramoto.pdf)\n\nCollective synchronization is quantified by the complex order parameter \\( Z = R \\cdot e^{i\\phi} = \\frac{1}{N}\\sum_{j=1}^{N} e^{i\\theta_{j}} \\). Its magnitude \\( R = |Z| \\) describes the level of synchronization: \\( R=1 \\) if and only if all oscillators are phase synchronized, and \\( R=0 \\) when phases are evenly distributed around the circle.[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) Below the critical coupling, \\( R \\) decays to \\( O(N^{-1/2}) \\) jitter; above it, \\( R \\) grows and saturates at a value below 1, with the population splitting into a frequency-locked cluster and drifting tails.[4](https://web.mit.edu/~jadbabai/www/ESE680/Strogatz_Kuramoto.pdf)\n\n## How it is done\n\nAnalysis proceeds through the self-consistency relation between \\( K \\) and \\( R \\). For a smooth even unimodal density \\( g(\\omega) \\), the critical coupling is \\( K_{c} = 2/[\\pi g(0)] \\), and \\( r = \\sqrt{1-K_{c}/K} \\) for all \\( K \\ge K_{c} \\), a formula later matched by numerical simulation.[4](https://web.mit.edu/~jadbabai/www/ESE680/Strogatz_Kuramoto.pdf) For Lorentzian frequencies of half-width \\( \\Delta \\), this becomes \\( K_{c} = 2\\Delta \\), with the incoherent state \\( R=0 \\) losing stability in a pitchfork bifurcation where \\( R = \\sqrt{1-2\\Delta/K} \\) becomes stable.[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) The bifurcation is supercritical if \\( g''(0) < 0 \\), the generic case for smooth unimodal even densities, and subcritical if \\( g''(0) > 0 \\).[4](https://web.mit.edu/~jadbabai/www/ESE680/Strogatz_Kuramoto.pdf) With white noise of strength \\( D \\), incoherence is linearly stable below the critical coupling (Lorentzian width \\( \\gamma \\)), recovering \\( K_{c} = 2/[\\pi g(0)] \\) at \\( D=0 \\).[1](https://doi.org/10.1103/revmodphys.77.137)\n\nDistribution shape controls the transition order. For bimodal \\( g(\\omega) \\) the eigenvalues may be complex, and different bifurcation scenarios and phase diagrams occur depending on the distribution[1](https://doi.org/10.1103/revmodphys.77.137); Kuramoto speculated that with a bimodal distribution the oscillators may form a single giant oscillator or disintegrate into two mutually unlocked crowds, depending on coupling strength, peak width, and spacing.[3](https://link.springer.com/article/10.1186/s13408-020-00086-9)\n\nFor Lorentzian \\( g(\\omega) \\), the Ott–Antonsen reduction yields the exact two-dimensional system \\( \\dot{Z} = (-\\Delta + i\\hat{\\omega}) \\cdot Z + \\frac{K}{2} \\cdot Z \\cdot (1-|Z|^{2}) \\); the reduction is exact, not approximate, and for multi-modal distributions there is one equation per mode.[3](https://link.springer.com/article/10.1186/s13408-020-00086-9)\n\n## Origin\n\nThe model was introduced in a 1975 note, \"Self-entrainment of a population of coupled non-linear oscillators\", which constructs a solvable model exhibiting mutual synchronization above a threshold coupling strength, with all-to-all coupling \\( V_{rs} = v/N \\) and a Lorentzian frequency distribution.[7](https://sci-hub.ru/storage/moscow/2116/b8b7875af4f0031b31a1cb79c1d884f6/selfentrainment-of-a-population-of-coupled-nonlinear-oscillators.pdf) The proposal was inspired by [Arthur T. Winfree](https://www.edgepedia.org/arthur-t-winfree)'s 1967 paper \"Biological Rhythms and the Behavior of Populations of Coupled Oscillators\"[8](https://doi.org/10.1016/0022-5193%2867%2990051-3), which described limit-cycle oscillators by phase only and claimed a critical condition for the onset of collective oscillation under global coupling.[9](https://google.iopscience.iop.org/article/10.1088/1742-5468/ae52eb) Finding Winfree's argument for the phase transition not convincing enough, Kuramoto modified the model into a mathematically tractable one, preserving the original form as much as possible, and handled it analytically via mean-field theory with a complex order parameter.[9](https://google.iopscience.iop.org/article/10.1088/1742-5468/ae52eb) The first survey of the model, by Juan A. Acebrón and colleagues, appeared in [Reviews of Modern Physics](https://www.edgepedia.org/reviews-of-modern-physics) in 2005.[1](https://doi.org/10.1103/revmodphys.77.137)\n\n## Variants\n\nSeveral named extensions broaden the original all-to-all, instantaneous, first-order form.\n\n**Phase lag.** A phase-lag (phase-frustration) parameter \\( \\alpha \\) generalizes the coupling to approximate a time delay in the interactions between oscillators.[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) Time-delayed coupling can be treated within the same reduction framework, where the delayed model yields an infinite-dimensional delay-differential equation for \\( r(t) \\).[10](https://ar5iv.labs.arxiv.org/html/0806.0004)\n\n**Networks.** Placing oscillators on graph nodes replaces the global mean field by a network Laplacian. For uncorrelated networks the critical coupling is \\( \\lambda_{c} = \\lambda_{\\max}/2 \\)[5](https://www.nature.com/articles/srep04783); on the fully connected graph it is rescaled by \\( \\langle k \\rangle/\\langle k^{2} \\rangle \\), so more heterogeneous networks synchronize at weaker coupling, and for \\( \\gamma = 3 \\) scale-free networks \\( \\lambda_{c} \\sim 1/\\ln N \\).[2](https://ar5iv.labs.arxiv.org/html/1511.07139)\n\n**Inertial (second-order) dynamics.** Adding inertia gives the second-order Kuramoto model, widely used for power grids and superconducting Josephson junctions; the Ott–Antonsen ansatz has been generalized to this setting on complex networks, reducing the ODE dimension from \\( N \\) to the number of possible degrees.[5](https://www.nature.com/articles/srep04783)\n\n**Higher-order interactions.** H. Daido introduced order parameters for macroscopic mutual entrainment in uniformly coupled limit-cycle oscillators in 1992.[11](https://doi.org/10.1143/ptp/88.6.1213) Takuma Tanaka and Toshio Aoyagi reported multistable attractors from three-body interactions in [Physical Review Letters](https://www.edgepedia.org/physical-review-letters) in 2011.[12](https://doi.org/10.1103/physrevlett.106.224101) Per Sebastian Skardal and Alex Arenas showed in Communications Physics in 2020 that higher-order interactions promote abrupt synchronization switching via a \\( (2,-1,-1) \\) coupling[13](https://doi.org/10.1038/s42005-020-00485-0), and in the same year Ana P. Millán, Joaquín J. Torres, and Ginestra Bianconi formulated a higher-order Kuramoto model with oscillators on nodes, links, and triangles, showing explosive transitions under adaptive coupling.[14](https://doi.org/10.1103/physrevlett.124.218301) Maxime Lucas, Giulia Cencetti, and Federico Battiston developed a multiorder Laplacian for arbitrary-order simplicial complexes in [Physical Review Research](https://www.edgepedia.org/physical-review-research) in 2020[15](https://doi.org/10.1103/physrevresearch.2.033410), and L. V. Gambuzza and colleagues analyzed stability of multi-order coupling in simplicial complexes in [Nature Communications](https://www.edgepedia.org/nature-communications) in 2021.[16](https://doi.org/10.1038/s41467-021-21486-9) The three-body interaction \\( \\sin(\\theta_{k}+\\theta_{l}-2\\theta_{j}) \\), the \\( (1,1,-2) \\) form, was rigorously derived from Stuart–Landau oscillators by Iván León and colleagues in Chaos in 2024.[17](https://doi.org/10.1063/5.0176748)\n\n## Applications\n\nIn neuroscience, individual Kuramoto oscillators can represent single neurons or large numbers of neurons in neural masses, with each population's macroscopic state given by its synchrony level \\( R_{\\sigma} \\) and average phase \\( \\phi_{\\sigma} \\).[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) The second-order model is a standard tool for power-grid stability, and basin stability has been applied to real grids.[2](https://ar5iv.labs.arxiv.org/html/1511.07139)\n\n## Limitations and alternatives\n\nThe model rests on [phase reduction](https://www.edgepedia.org/phase-reduction), which requires weak coupling (\\( \\kappa \\ll 1 \\)), nearly identical oscillators, and pairwise coupling; the shape of the phase interaction function is decisive for collective behavior.[18](https://bastianpietras.github.io/files/2019_Pietras_Daffertshofer.pdf) Mean-field reduction methods assume a coupling function with a single harmonic, and explicit examples show the reductions become invalid, with chaotic dynamics occurring where the reduction would yield an effective two-dimensional phase space.[3](https://link.springer.com/article/10.1186/s13408-020-00086-9) Kuramoto's calculation of the partially synchronized phase does not indicate whether that phase is stable, globally or even locally[4](https://web.mit.edu/~jadbabai/www/ESE680/Strogatz_Kuramoto.pdf), while rigorous analyses exist: Chiba proved that the incoherent solution is linearly stable below Kuramoto's transition point \\( K_{c} \\) and unstable above it, for arbitrary frequency distributions, and later work established asymptotic stability of the incoherent state with [bifurcation analysis](https://www.edgepedia.org/bifurcation-analysis); below \\( K_{c} \\) the incoherent state is only neutrally stable, with disturbances decaying like [Landau damping](https://www.edgepedia.org/landau-damping) in plasmas.[1](https://doi.org/10.1103/revmodphys.77.137)\n\nOn heterogeneous networks the model has no exact solution, because the equations cannot be decoupled by a global mean field. For scale-free networks with \\( 2 \\le \\gamma \\le 3 \\), the mean-field approximation predicts a vanishing \\( \\lambda_{c} \\), but simulations show \\( \\lambda_{c} \\) converging to a constant; the source of this disagreement remains an open problem.[2](https://ar5iv.labs.arxiv.org/html/1511.07139) The order parameter \\( r \\) is also less useful for models with short-range coupling, where more complex synchronization situations arise.[1](https://doi.org/10.1103/revmodphys.77.137)\n\nCompared with the Winfree model, which incorporates separate influence and sensitivity functions, the Kuramoto model can be viewed as a short-time approximation in the small natural frequency and coupling strength regime, but their large-time dynamics are completely different: the Kuramoto model has a balance of total phase, which excludes the complete oscillator death seen in Winfree models.[19](https://ems.press/content/serial-article-files/36988) The Ott–Antonsen reduction allows nonidentical oscillators but requires Lorentzian-type distributions with support over all real frequencies, whereas the Watanabe–Strogatz reduction of finite networks requires identical oscillators; neither applies to finite networks of nonidentical oscillators.[3](https://link.springer.com/article/10.1186/s13408-020-00086-9)\n\n## References\n\n1. [Juan A. Acebrón and colleagues (2005). The Kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics.](https://doi.org/10.1103/revmodphys.77.137)\n2. [The Kuramoto model in complex networks (arXiv:1511.07139 review)](https://ar5iv.labs.arxiv.org/html/1511.07139)\n3. [Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review (J. Math. Neurosci., 2020)](https://link.springer.com/article/10.1186/s13408-020-00086-9)\n4. [From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators (Strogatz, Physica D 143, 2000)](https://web.mit.edu/~jadbabai/www/ESE680/Strogatz_Kuramoto.pdf)\n5. [Low-dimensional behavior of Kuramoto model with inertia in complex networks (Scientific Reports, 2014)](https://www.nature.com/articles/srep04783)\n6. [The mathematics of asymptotic stability in the Kuramoto model (Proc. R. Soc. A)](https://royalsocietypublishing.org/rspa/article/474/2220/20180467/80056/The-mathematics-of-asymptotic-stability-in-the)\n7. [Self-entrainment of a population of coupled non-linear oscillators](https://sci-hub.ru/storage/moscow/2116/b8b7875af4f0031b31a1cb79c1d884f6/selfentrainment-of-a-population-of-coupled-nonlinear-oscillators.pdf)\n8. [Biological rhythms and the behavior of populations of coupled oscillators (Journal of Theoretical Biology, 1967)](https://doi.org/10.1016/0022-5193%2867%2990051-3)\n9. [Half a century of the theory of synchronization (Y. Kuramoto, J. Stat. Mech. (2026) 044001)](https://google.iopscience.iop.org/article/10.1088/1742-5468/ae52eb)\n10. [Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators (Ott & Antonsen-type reduction, arXiv:0806.0004)](https://ar5iv.labs.arxiv.org/html/0806.0004)\n11. [H. Daido (1992). Order Function and Macroscopic Mutual Entrainment in Uniformly Coupled Limit-Cycle Oscillators. Progress of Theoretical Physics.](https://doi.org/10.1143/ptp/88.6.1213)\n12. [Takuma Tanaka, Toshio Aoyagi (2011). Multistable Attractors in a Network of Phase Oscillators with Three-Body Interactions. Physical Review Letters.](https://doi.org/10.1103/physrevlett.106.224101)\n13. [Per Sebastian Skardal, Alex Arenas (2020). Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching. Communications Physics.](https://doi.org/10.1038/s42005-020-00485-0)\n14. [Ana P. Millán, Joaquín J. Torres, Ginestra Bianconi (2020). Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes. Physical Review Letters.](https://doi.org/10.1103/physrevlett.124.218301)\n15. [Maxime Lucas, Giulia Cencetti, Federico Battiston (2020). Multiorder Laplacian for synchronization in higher-order networks. Physical Review Research.](https://doi.org/10.1103/physrevresearch.2.033410)\n16. [L. V. Gambuzza and colleagues (2021). Stability of synchronization in simplicial complexes. Nature Communications.](https://doi.org/10.1038/s41467-021-21486-9)\n17. [Iván León and colleagues (2024). Higher-order interactions induce anomalous transitions to synchrony. Chaos An Interdisciplinary Journal of Nonlinear Science.](https://doi.org/10.1063/5.0176748)\n18. [Network dynamics of coupled oscillators: phase reduction theory (Pietras & Daffertshofer, Physics Reports 819, 2019)](https://bastianpietras.github.io/files/2019_Pietras_Daffertshofer.pdf)\n19. [Collective synchronization of classical and quantum oscillators (EMS Surveys)](https://ems.press/content/serial-article-files/36988)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Dynamical systems, chaos, and ergodic theory*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 21, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://web.mit.edu/~jadbabai/www/ESE680/Strogatz_Kuramoto.pdf" ], "url": "https://www.edgepedia.org/kuramoto-model", "markdown_url": "https://www.edgepedia.org/kuramoto-model.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Kuramoto model\", Edgepedia (EdgeChat), https://www.edgepedia.org/kuramoto-model. Edgepedia Community License 1.0.", "credit_md": "\"[Kuramoto model](https://www.edgepedia.org/kuramoto-model)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/kuramoto-model](https://www.edgepedia.org/kuramoto-model). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Kuramoto model\", Edgepedia (EdgeChat), https://www.edgepedia.org/kuramoto-model. Edgepedia Community License 1.0.", "speakable": "The Kuramoto model, introduced by Yoshiki Kuramoto in 1975, is a mathematical model of synchronization in which coupled phase oscillators entrain into collective oscillation above a critical coupling strength." }