What is nonlinear science?
Stanislaw Ulam reportedly said (something like) "Calling a science 'nonlinea
r' is like calling zoology 'the study of non-human animals'. So why do we ha
ve a name that appears to be merely a negative?
Firstly, linearity is rather special, and no model of a real system is truly
linear (you might protest that quantum mechanics is an exception, however t
his is at the expense of infinite dimensionality which is just as bad or wor
se-- and 'any' finite dimensional nonlinear model can be turned into an infi
nite dimensional linear one--e.g. a map x'=f(x) is equivalent to the linear
integral equation
p'(x) = integral [ p(y) \delta(x-f(y)) dy ]).
Some things are profitably studied as linear approximations to the real mode
ls--for example the fact that Hooke's law, the linear law of elasticity (str
ain is proportional to stress) is approximately valid for a pendulum of smal
l amplitude implies that its period is approximately independent of amplitud
e. However, as the amplitude gets large the period gets longer, a fundamenta
l effect of nonlinearity in the pendulum equations (see <http://monet.physik
.unibas.ch/~elmer/pendulum/upend.htm> and [3.10]).
Secondly, nonlinear systems have been shown to exhibit surprising and comple
x effects that would never be anticipated by a scientist trained only in lin
ear techniques. Prominent examples of these include bifurcation, chaos, and
solitons. Nonlinearity has its most profound effects on dynamical systems (s
ee [2.3]).
Further, while we can enumerate the linear objects, nonlinear ones are nonde
numerable, and as of yet mostly unclassified. We currently have no general t
echniques (and very few special ones) for telling whether a particular nonli
near system will exhibit the complexity of chaos, or the simplicity of order
. Thus since we cannot yet subdivide nonlinear science into proper subfields
, it exists has a whole.
Nonlinear science has applications to a wide variety of fields, from mathema
tics, physics, biology, and chemistry, to engineering, economics, and medici
ne. This is one of its most exciting aspects--that it brings researchers fro
m many disciplines together with a common language.
--
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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※ 發信站: 批踢踢實業坊(ptt.csie.ntu.edu.tw)
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