This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

Stanislaw Ulam reportedly said (something like) "Calling a science 'nonlinear'

is like calling zoology 'the study of non-human animals'. So why do we have a

name that appears to be merely a negative?

Firstly, linearity is rather special, and no model of a real system is truly

linear. Some things are profitably studied as linear approximations to the

real models--for example the fact that Hooke's law, the linear law of

elasticity (strain is proportional to stress) is approximately valid for a

pendulum of small amplitude implies that its period is approximately

independent of amplitude. However, as the amplitude gets large the period gets

longer, a fundamental effect of nonlinearity in the pendulum equations (see

http://monet.physik.unibas.ch/~elmer/pendulum/upend.htm and [3.10]).

(You might protest that quantum mechanics is the fundamental theory and that

it is linear! However this is at the expense of infinite dimensionality which

is just as bad or worse--and 'any' finite dimensional nonlinear model can be

turned into an infinite dimensional linear one--e.g. a map x' = f(x) is

equivalent to the linear integral equation often called the Perron-Frobenius

equation

p'(x) = integral [ p(y) \delta(x-f(y)) dy ])

Here p(x) is a density, which could be interpreted as the probability of

finding oneself at the point x, and the Dirac-delta function effectively moves

the points according to the map f to give the new density. So even a nonlinear

map is equivalent to a linear operator.)

Secondly, nonlinear systems have been shown to exhibit surprising and complex

effects that would never be anticipated by a scientist trained only in linear

techniques. Prominent examples of these include bifurcation, chaos, and

solitons. Nonlinearity has its most profound effects on dynamical systems (see

[2.3]).

Further, while we can enumerate the linear objects, nonlinear ones are

nondenumerable, and as of yet mostly unclassified. We currently have no

general techniques (and very few special ones) for telling whether a

particular nonlinear 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 as a whole.

Nonlinear science has applications to a wide variety of fields, from

mathematics, physics, biology, and chemistry, to engineering, economics, and

medicine. This is one of its most exciting aspects--that it brings researchers

from many disciplines together with a common language.

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