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  "body": "\n## Example\n\nLet $K \\in \\R$ be a parameter.\nThe *standard map*, or Chirikov–Taylor map, acts on points $(\\theta, p)$ of the cylinder, where $\\theta$ is taken modulo $2\\pi$.\nOn the torus, $p$ is also taken modulo $2\\pi$.\nThe map is\n$$p_{n+1} = p_n + K \\sin \\theta_n, \\qquad \\theta_{n+1} = \\theta_n + p_{n+1}.$$\nIts Jacobian matrix with respect to $(p_n, \\theta_n)$ is\n$$\\begin{pmatrix} 1 \u0026 K \\cos \\theta_n \\\\ 1 \u0026 1 + K \\cos \\theta_n \\end{pmatrix},$$\nwhose determinant is $1 + K \\cos \\theta_n - K \\cos \\theta_n = 1$.\nSo the map preserves area and orientation, and it is a [[symplectomorphism|A8TS2SNX9Z]] of the plane form $dp \\wedge d\\theta$.\n\n## Remarks\n\nFor $K = 0$ the map is integrable: $p$ is constant and each [[circle|060W5CBZP6]] $p = c$ is invariant, with rotation by the angle $c$.\nFor small $K \\ne 0$, [[KAM theory|3ZZSXKM16Z]] for twist maps shows that the invariant circles with sufficiently irrational rotation numbers survive, slightly deformed.\nThese circles wrap around the cylinder and prevent $p$ from growing without bound.\nAs $K$ grows the circles break; numerically, the last one disappears near $K \\approx 0.9716$, found by Greene, and beyond that $p$ can drift without bound.\nFor large $K$ most orbits look [[chaotic|DBKXCGDKTD]], with islands of stable motion around periodic orbits.\n\nThe map models a kicked rotor: a rotor that receives a periodic kick whose strength depends on its angle.\n\n## Explanations\n\n### L1 human\n\nThe standard map is a simple rule that takes a point of a cylinder to a new point, with one parameter $K$.\nFor $K = 0$ nothing interesting happens: each point slides along its circle.\nAs $K$ grows, the picture of its orbits changes from neat curves to a mixture of curves, islands and chaotic seas.\nIt is the standard laboratory for seeing the transition from order to chaos in Hamiltonian systems.\n",
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