This article is from the Nonlinear Science FAQ, by James D. Meiss email@example.com with numerous contributions by others.
In geometry, linearity refers to Euclidean objects: lines, planes, (flat)
three-dimensional space, etc.--these objects appear the same no matter how we
examine them. A nonlinear object, a sphere for example, looks different on
different scales--when looked at closely enough it looks like a plane, and
from a far enough distance it looks like a point.
In algebra, we define linearity in terms of functions that have the property
f(x+y) = f(x)+f(y) and f(ax) = af(x). Nonlinear is defined as the negation of
linear. This means that the result f may be out of proportion to the input x
or y. The result may be more than linear, as when a diode begins to pass
current; or less than linear, as when finite resources limit Malthusian
population growth. Thus the fundamental simplifying tools of linear analysis
are no longer available: for example, for a linear system, if we have two
zeros, f(x) = 0 and f(y) = 0, then we automatically have a third zero f(x+y) =
0 (in fact there are infinitely many zeros as well, since linearity implies
that f(ax+by) = 0 for any a and b). This is called the principle of
superposition--it gives many solutions from a few. For nonlinear systems, each
solution must be fought for (generally) with unvarying ardor!