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  "body": "\n## Definition\n\nLet $f \\colon X \\to X$ be a continuous map of a metric space, or a flow on a metric space, and let $A$ be an [[attractor|7FWX24X5MW]] of it.\nThe attractor $A$ is *strange* if it has a fractal structure.\nIn the most common precise form, this means that the [[Hausdorff dimension|SSNFTHZ6YJ]] of $A$, or its box-counting dimension, is not an integer.\n\n## Remarks\n\nThere is no single agreed definition.\nMany authors use strange attractor and chaotic attractor as synonyms, and then they ask for sensitive dependence on initial conditions on $A$ instead of, or in addition to, fractal structure.\nThe two conditions are different: there are chaotic attractors that are not fractal, and there are fractal attractors on which nearby orbits do not separate.\n\nExample: the Smale solenoid map doubles the angle $\\theta \\mapsto 2\\theta$ around a solid torus and contracts strongly in the two transverse directions.\nIts attractor is locally the product of an interval and a Cantor set.\nSo its dimension lies strictly between $1$ and $2$.\n\nNon-example: for the flow of $r' = r(1 - r^2)$, $\\theta' = 1$ in polar coordinates in the plane, the unit circle is an attractor.\nIt is a smooth closed curve of Hausdorff dimension $1$, an integer, so it is not strange.\n",
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