 # 2.2] What is nonlinear science?

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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