[2.10] What is sensitive dependence on initial conditions? Consider a boulder precariously perched on the top of an ideal hill. The sli ghtest push will cause the boulder to roll down one side of the hill or the other: the subsequent behavior depends sensitively on the direction of the p ush--and the push can be arbitrarily small. Of course, it is of great import ance to you which direction the boulder will go if you are standing at the b ottom of the hill on one side or the other! Sensitive dependence is the equivalent behavior for every initial condition- -every point in the phase space is effectively perched on the top of a hill. More precisely a set S exhibits sensitive dependence if there is an r such t hat for any epsilon > 0 and for each x in S, there is a y such that |x - y| < epsilon, and |x_n - y_n| > r for some n > 0. Then there is a fixed distanc e r (say 1), such that no matter how precisely one specifies an initial stat e there are nearby states that eventually get a distance r away. Note: sensitive dependence does not require exponential growth of perturbati ons (positive Lyapunov exponent), but this is typical (see [2.14]) for chaot ic systems. Note also that we most definitely do not require ALL nearby init ial points diverge--generically [2.14] this does not happen--some nearby poi nts may converge. (We may modify our hilltop analogy slightly and say that e very point in phase space acts like a high mountain pass.) Finally, the word s "initial conditions" are a bit misleading: a typical small disturbance int roduced at any time will grow similarly. Think of "initial" as meaning "a ti me when a disturbance or error is introduced," not necessarily time zero. -- 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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