[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). -- 在细雨的午後 书页里悉哩哩地传来 " 周期3 = ? " 然而我知道 当我正在日耳曼深处的黑森林 继续发掘海森堡未曾做过的梦时 康德的诺言早已远离......... 远来的传教士静静地看着山涧不断反覆叠代自己的 过去 现在 和 未来 於是仅以 一颗量子浑沌 一本符号动力学 祝那发生在周一下午的新生 --

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