[2.8] What is an attractor? Informally an attractor is simply a state into which a system settles (thus dissipation is needed). Thus in the long term, a dissipative dynamical syste m may settle into an attractor. Interestingly enough, there is still some controversy in the mathematics com munity as to an appropriate definition of this term. Most people adopt the d efinition Attractor: A set in the phase space that has a neighborhood in which every p oint stays nearby and approaches the attractor as time goes to infinity. Thus imagine a ball rolling inside of a bowl. If we start the ball at a poin t in the bowl with a velocity too small to reach the edge of the bowl, then eventually the ball will settle down to the bottom of the bowl with zero vel ocity: thus this equilibrium point is an attractor. The neighborhood of poin ts that eventually approach the attractor is the basin of attraction for the attractor. In our example the basin is the set of all configurations corres ponding to the ball in the bowl, and for each such point all small enough ve locities (it is a set in the four dimensional phase space [2.4]). Attractors can be simple, as the previous example. Another example of an att ractor is a limit cycle, which is a periodic orbit that is attracting (limit cycles can also be repelling). More surprisingly, attractors can be chaotic (see [2.9]) and/or strange (see [2.12]). The boundary of a basin of attraction is often a very interesting object sin ce it distinguishes between different types of motion. Typically a basin bou ndary is a saddle orbit, or such an orbit and its stable manifold. A crisis is the change in an attractor when its basin boundary is destroyed. An alternative definition of attractor is sometimes used because there are s ystems that have sets that attract most, but not all, initial conditions in their neighborhood (such phenomena is sometimes called riddling of the basin ). Thus, Milnor defines an attractor as a set for which a positive measure ( probability, if you like) of initial conditions in a neighborhood are asympt otic to the set. -- 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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