[2.12] What is a Strange Attractor? Before Chaos (BC?), the only known attractors (see [2.8]) were fixed points, periodic orbits (limit cycles), and invariant tori (quasiperiodic orbits). In fact the famous Poincar□-Bendixson theorem states that for a pair of fir st order differential equations, only fixed points and limit cycles can occu r (there is no chaos in 2D flows). In a famous paper in 1963, Ed Lorenz discovered that simple systems of three differential equations can have complicated attractors. The Lorenz attracto r (with its butterfly wings reminding us of sensitive dependence (see [2.10] )) is the "icon" of chaos <http://kong.apmaths.uwo.ca/~bfraser/version1/lore nzintro.html>. Lorenz showed that his attractor was chaotic, since it exhibi ted sensitive dependence. Moreover, his attractor is also "strange," which m eans that it is a fractal (see [3.2]). The term strange attractor was introduced by Ruelle and Takens in 1970 in th eir discussion of a scenario for the onset of turbulence in fluid flow. They noted that when periodic motion goes unstable (with three or more modes), t he typical (see [2.14]) result will be a geometrically strange object. Unfortunately, the term strange attractor is often used for any chaotic attr actor. However, the term should be reserved for attractors that are "geometr ically" strange, e.g. fractal. One can have chaotic attractors that are not strange (a trivial example would be to take a system like the cat map, which has the whole plane as a chaotic set, and add a third dimension which is si mply contracting onto the plane). There are also strange, nonchaotic attract ors (see Grebogi, C., et al. (1984). "Strange Attractors that are not Chaoti c." Physica D 13: 261-268). -- 在細雨的午後 書頁裡悉哩哩地傳來 " 週期3 = ? " 然而我知道 當我正在日耳曼深處的黑森林 繼續發掘海森堡未曾做過的夢時 康德的諾言早已遠離......... 遠來的傳教士靜靜地看著山澗不斷反覆疊代自己的 過去 現在 和 未來 於是僅以 一顆量子渾沌 一本符號動力學 祝那發生在週一下午的新生 --

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