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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