Nonlinear system

In physics and other sciences, a nonlinear system, in contrast to a linear system, is a system which does not satisfy the superposition principle – meaning that the output of a nonlinear system is not directly proportional to the input.

Description

In mathematics, a nonlinear system of equations is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.

In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. It does not matter if nonlinear known functions appear in the equations.

In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.

Typically, the behavior of a nonlinear system is described by a nonlinear system of equations. Nonlinear problems are of interest to engineers, physicists and mathematicians and many other scientists because most systems are inherently nonlinear in nature.

As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as chaos and singularities are hidden by linearization.

It follows that some aspects of the behavior of a nonlinear system appear commonly to be chaotic, unpredictable or counterintuitive. Although such chaotic behavior may resemble random behavior, it is absolutely not random.

For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.

Terminology

Some authors use the term nonlinear science for the study of nonlinear systems. This is disputed by others:

Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.

— Stanislaw Ulam

See also

• Aleksandr Mikhailovich Lyapunov
• Chaos theory
• Dynamical system
• Initial condition
• Interaction
• Linear system
• Mode coupling
• Superposition principle
• Vector soliton
• Volterra series

External links

• Nonlinear systems @ Wikipedia