[2.9] What is chaos?
It has been said that "Chaos is a name for any order that produces confusion
in our minds." (George Santayana, thanks to Fred Klingener for finding this
). However, the mathematical definition is, roughly speaking,
Chaos: effectively unpredictable long time behavior arising in a determinist
ic dynamical system because of sensitivity to initial conditions.
It must be emphasized that a deterministic dynamical system is perfectly pre
dictable given perfect knowledge of the initial condition, and is in practic
e always predictable in the short term. The key to long-term unpredictabilit
y is a property known as sensitivity to (or sensitive dependence on) initial
conditions.
For a dynamical system to be chaotic it must have a 'large' set of initial c
onditions which are highly unstable. No matter how precisely you measure the
initial condition in these systems, your prediction of its subsequent motio
n goes radically wrong after a short time. Typically (see [2.14] for one def
inition of 'typical'), the predictability horizon grows only logarithmically
with the precision of measurement (for positive Lyapunov exponents, see [2.
11]). Thus for each increase in precision by a factor of 10, say, you may on
ly be able to predict two more time units (measured in units of the Lyapunov
time, i.e. the inverse of the Lyapunov exponent).
More precisely: A map f is chaotic on a compact invariant set S if
(i) f is transitive on S (there is a point x whose orbit is dense in S), and
(ii) f exhibits sensitive dependence on S (see [2.10]).
To these two requirements Devaney adds the requirement that periodic points
are dense in S, but this doesn't seem to be really in the spirit of the noti
on, and is probably better treated as a theorem (very difficult and very imp
ortant), and not part of the definition.
Usually we would like the set S to be a large set. It is too much to hope fo
r except in special examples that S be the entire phase space. If the dynami
cal system is dissipative then we hope that S is an attractor (see [2.8]) wi
th a large basin. However, this need not be the case--we can have a chaotic
saddle, an orbit that has some unstable directions as well as stable directi
ons.
As a consequence of long-term unpredictability, time series from chaotic sys
tems may appear irregular and disorderly. However, chaos is definitely not (
as the name might suggest) complete disorder; it is disorder in a determinis
tic dynamical system, which is always predictable for short times.
The notion of chaos seems to conflict with that attributed to Laplace: given
precise knowledge of the initial conditions, it should be possible to predi
ct the future of the universe. However, Laplace's dictum is certainly true f
or any deterministic system, recall [2.3]. The main consequence of chaotic m
otion is that given imperfect knowledge, the predictability horizon in a det
erministic system is much shorter than one might expect, due to the exponent
ial growth of errors. The belief that small errors should have small consequ
ences was perhaps engendered by the success of Newton's mechanics applied to
planetary motions. Though these happen to be regular on human historic time
scales, they are chaotic on the 5 million year time scale (see e.g. "Newton
's Clock", by Ivars Peterson (1993 W.H. Freeman).
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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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