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  "body": "\n## Example\n\nLet $N \\ge 2$, let $K \\ge 0$ be a coupling strength, and let $\\omega_1, \\ldots, \\omega_N \\in \\R$ be natural frequencies.\nThe *Kuramoto model* is the [[autonomous system|0673D85QKZ]] for phases $\\theta_1, \\ldots, \\theta_N \\in \\R / 2\\pi\\Z$:\n$$\\theta_i' = \\omega_i + \\frac{K}{N} \\sum_{j=1}^{N} \\sin(\\theta_j - \\theta_i), \\qquad i = 1, \\ldots, N.$$\nEach oscillator turns at its own speed and is pulled towards the phases of the others.\n\nFor $N = 2$ the phase difference $\\psi = \\theta_2 - \\theta_1$ satisfies\n$$\\psi' = (\\omega_2 - \\omega_1) - K \\sin \\psi.$$\nIf $K \u003e 0$ and $K \\ge \\abs{\\omega_2 - \\omega_1}$, this equation has an equilibrium $\\psi^*$ with $\\sin \\psi^* = (\\omega_2 - \\omega_1)/K$.\nThen the two oscillators lock their phases at the difference $\\psi^*$ and turn at the common speed $(\\omega_1 + \\omega_2)/2$.\nIf $K \u003c \\abs{\\omega_2 - \\omega_1}$, then $\\psi' \\ne 0$ always, and the phases drift apart.\n\n## Remarks\n\nThe degree of synchrony is measured by the order parameter $r e^{i\\phi} = \\frac{1}{N} \\sum_j e^{i \\theta_j}$, with $0 \\le r \\le 1$.\nWith it the equations become $\\theta_i' = \\omega_i + K r \\sin(\\phi - \\theta_i)$, so each oscillator couples only to the mean field.\nIn the limit $N \\to \\infty$ with frequencies drawn from a symmetric unimodal density $g$, a partially synchronized state with $r \u003e 0$ appears when $K$ exceeds $K_c = 2 / (\\pi g(0))$.\nFor the Lorentzian density $g(\\omega) = \\gamma / (\\pi(\\gamma^2 + \\omega^2))$ this gives $K_c = 2\\gamma$, and the stationary order parameter is $r = \\sqrt{1 - K_c/K}$ for $K \u003e K_c$.\n\nKuramoto introduced the model in 1975.\nIt is the basic model of [[synchronization|YP1PMW0HE3]] of oscillators in physics, chemistry and biology.\n\n## Explanations\n\n### L1 human\n\nPicture many runners on a circular track, each with a favourite speed, and each slightly attracted to the others.\nIf the attraction is weak, they spread out and run independently.\nIf it is strong enough, a group forms that runs together, and the group grows as the attraction increases.\nThe Kuramoto model is the simplest equation that shows this sudden onset of collective rhythm, seen in fireflies flashing together and in pacemaker cells of the heart.\n",
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