[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
--
※ 發信站: 批踢踢實業坊(ptt.csie.ntu.edu.tw)
◆ From: 140.112.102.146