# 2.1] What is nonlinear?

## Description

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

# 2.1] What is nonlinear?

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!

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