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  "body": "\n## Summary\n\nThe circle map, or standard circle map, is the two-parameter family $\\theta \\mapsto \\theta + \\Omega - \\frac{K}{2\\pi} \\sin(2\\pi\\theta)$ of maps of the [[circle|060W5CBZP6]] $\\R/\\Z$.\nIt models a rotation by $\\Omega$ disturbed by a nonlinear kick of strength $K$.\nIt is the basic model of mode locking: its [[rotation number|DP4WWQBJJF]] locks to rational values on the [[Arnold tongues|WQF5ASXZTH]].\nFor $0 \\le K \u003c 1$ it is a family of circle [[diffeomorphisms|0KZZ70ZHXP]], and at $K = 1$ it has a critical point.\n\n## Explanations\n\n### L1 human\n\nPicture a point moving around a circle in steps.\nWithout the kick ($K = 0$), each step is the same turn by $\\Omega$.\nThe kick speeds the point up in some places and slows it down in others.\nA pendulum clock pushed by a periodic force, or a heart driven by a pacemaker, can be modelled this way.\nThe question is whether the motion locks to the rhythm of the drive.\n",
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