作者Keelungman (2000大跃进)
看板NTUNL
标题[2.10] What is sensitive dependence on initial conditions?
时间Tue Oct 2 12:21:39 2001
[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.
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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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