Nonlinear System - Definition

Definition

In mathematics, a linear function (or map) is one which satisfies both of the following properties:

  • additivity,
  • homogeneity,

(Additivity implies homogeneity for any rational α, and, for continuous functions, for any real α. For a complex α, homogeneity does not follow from additivity; for example, an antilinear map is additive but not homogeneous.) The conditions of additivity and homogeneity are often combined in the superposition principle

An equation written as

is called linear if is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if .

The definition is very general in that can be any sensible mathematical object (number, vector, function, etc.), and the function can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If contains differentiation of, the result will be a differential equation.

Read more about this topic:  Nonlinear System

Famous quotes containing the word definition:

    ... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.
    George Eliot [Mary Ann (or Marian)

    I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.”
    Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.
    Jean Baudrillard (b. 1929)