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:

    Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.
    Elaine Heffner (20th century)

    ... 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)

    According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.
    Ana Castillo (b. 1953)