# User Contributed Dictionary

# Extensive Definition

- This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity (disambiguation).

In mathematics, a nonlinear
system is a system which is not linear,
i.e. a system which does not satisfy the superposition
principle. Less technically, a nonlinear system is any problem
where the variable(s) to be solved for cannot be written as a
linear sum of independent components. A nonhomogenous system, which
is linear apart from the presence of a function of the independent
variables, is nonlinear according to a strict definition, but
such systems are usually studied alongside linear systems, because
they can be transformed to a linear system as long as a particular
solution is known.

Generally, nonlinear problems are difficult (if
possible) to solve and are much less understandable than linear
problems. Even if not exactly solvable, the outcome of a linear
problem is rather predictable, while the outcome of a nonlinear is
inherently not.

Nonlinear problems are of interest to physicists and mathematicians because
most physical systems are inherently nonlinear in nature. Physical
examples of linear systems are not very common. Nonlinear equations
are difficult to solve and give rise to interesting phenomena such
as chaos. The
weather is famously nonlinear, where simple changes in one part of
the system produce complex effects throughout.

## Definition

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

- Additivity: f(x + y) = f(x) + f(y)\,
- Homogeneity: f(\alpha x) = \alpha f(x)\,

An equation written as

- f(x) = C\,

is called linear if f(x) is linear (as
defined above) and nonlinear otherwise. Note that x does not need
to be a scalar (can be a vector, function,
etc), and that C must not depend on x. The equation is called
homogeneous if C = 0.

## Nonlinear algebraic equations

Generally, nonlinear algebraic problems are often
exactly solvable, and if not they usually can be thoroughly
understood through qualitative
and numeric
analysis. As an example, the equation

- x^2 + x - 1 = 0\,

may be written as

- f(x) = C \quad \mbox \quad f(x) = x^2 + x \quad \mbox \quad C = 1\,

and is nonlinear because f(x) satisfies neither
additivity nor
homogeneity (the
nonlinearity is due to the x^2). Though nonlinear, this simple
example may be solved exactly (via the quadratic
formula) and is very well understood. On the other hand, the
nonlinear equation

- x^5 - x - 1 = 0\,

is not exactly solvable (see quintic
equation), though it may be qualitatively analyzed and is well
understood, for example through making a graph and examining the
roots
of f(x) - C = 0.

## Nonlinear recurrence relations

A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences.## Nonlinear differential equations

Problems involving nonlinear differential
equations are extremely diverse, and methods of solution or
analysis are very problem dependent.

One of the greatest difficulties of nonlinear
problems is that it is not generally possible to combine known
solutions into new solutions. In linear problems, for example, a
family of linearly
independent solutions can be used to construct general
solutions through the superposition
principle. A good example of this is one-dimensional heat
transport with
Dirichlet boundary conditions, the solution of which can be
written as a time-dependent linear combination of sinusoids of
differing frequencies, this makes solutions very flexible. It is
often possible to find several very specific solutions to nonlinear
equations, however the lack of a superposition
principle prevents the construction of new solutions.

### Ordinary differential equations

First order
ordinary differential equations are often exactly solvable by
separation
of variables, especially for autonomous equations. For example,
the nonlinear equation

- \frac = -u^2\,

will easily yield u = (x + C)^ as a general
solution which happens to be simpler than the solution to the
linear equation du/dx = -u. The equation is nonlinear because it
may be written as

- \frac + u^2=0\,

and the left-hand side of the equation is not a
linear function of u and its derivatives. Note that if the u² term
were replaced with u, the problem would be linear (the exponential
decay problem).

Second and higher order ordinary differential
equations (more generally, systems of nonlinear equations) rarely
yield closed form
solutions, though implicit solutions and solutions involving
nonelementary
integrals are encountered.

Common methods for the qualitative analysis of
nonlinear ordinary differential equations include:

- Examination of any conserved quantities, especially in Hamiltonian systems.
- Examination of dissipative quantities (see Lyapunov function) analogous to conserved quantities.
- Linearization via Taylor expansion.
- Change of variables into something easier to study.
- Bifurcation theory.
- Perturbation methods (can be applied to algebraic equations too).

### Partial differential equations

The most common basic approach to studying
nonlinear
partial differential equations is to change the variables (or
otherwise transform the problem) so that the resulting problem is
simpler (possibly even linear). Sometimes, the equation may be
transformed into one or more
ordinary differential equations, as seen in the similarity
transform or separation
of variables, which is always useful whether or not the
resulting ordinary differential equation(s) is solvable.

Another common (though less mathematic) tactic,
often seen in fluid and heat mechanics, is to use
scale analysis to simplify a general, natural equation in a
certain specific boundary
value problem. For example, the (very) nonlinear Navier-Stokes
equations can be simplified into one linear partial
differential equation in the case of transient, laminar, one
dimensional flow in a circular pipe; the scale analysis provides
conditions under which the flow is laminar and one dimensional and
also yields the simplified equation.

Other methods include examining the characteristics
and using the methods outlined above for ordinary differential
equations.

### Example: pendulum

A classic, extensively studied nonlinear problem
is the dynamics of a pendulum.
Using Lagrangian
mechanics, it may be shown that the motion of a pendulum can be
described by the dimensionless nonlinear
equation

- \frac + \sin(\theta) = 0\,

where gravity is "down" and \theta is as shown in
the figure at right. One approach to "solving" this equation is to
use \scriptstyle \frac as an integrating
factor, which would eventually yield

- \int \frac = t + C_1\,

which is an implicit solution involving an
elliptic
integral. This "solution" generally does not have many uses
because most of the nature of the solution is hidden in the
nonelementary integral (nonelementary even if C_0 = 0).

Another way to approach the problem is to
linearize any nonlinearities (the sine function term in this case)
at the various points of interest through Taylor
expansions. For example, the linearization at \theta = 0,
called the small angle approximation, is

- \frac + \theta = 0\,

since \sin(\theta) \approx \theta for \theta
\approx 0. This is a simple
harmonic oscillator corresponding to oscillations of the
pendulum near the bottom of its path. Another linearization would
be at \theta = \pi, corresponding to the pendulum being straight
up:

- \frac + \pi - \theta = 0\,

since \sin(\theta) \approx \pi - \theta for
\theta \approx \pi. The solution to this problem involves hyperbolic
sinusoids, and note that unlike the small angle approximation,
this approximation is unstable, meaning that |\theta| will usually
grow without limit, though bounded solutions are possible. This
corresponds to the difficulty of balancing a pendulum upright, it
is literally an unstable state.

One more interesting linearization is possible
around \theta = \pi/2, around which \sin(\theta) \approx 1:

- \frac + 1 = 0.

This corresponds to a free fall problem. A very
useful qualitative picture of the pendulum's dynamics may be
obtained by piecing together such linearizations, as seen in the
figure at right. Other techniques may be used to find (exact)
phase
portraits and approximate periods.

## Metaphorical use

Engineers often use the term nonlinear to refer
to irrational behavior, with the implication that the person who
has become nonlinear is on the edge of losing control or even
having a nervous
breakdown.

## Types of nonlinear behaviors

- Indeterminism - the behavior of a system cannot be predicted.
- Multistability - alternating between two or more exclusive states.
- Aperiodic oscillations - functions that do not repeat values after some period (otherwise known as chaotic oscillations or chaos).

## Examples of nonlinear equations

- AC power flow model
- Bellman equation for optimal policy
- Boltzmann transport equation
- General relativity
- Ginzburg-Landau equation
- Navier-Stokes equations of fluid dynamics
- Korteweg–de Vries equation
- nonlinear optics
- nonlinear Schrödinger equation
- Richards equation for unsaturated water flow
- Robot unicycle balancing
- Sine-Gordon equation
- Landau-Lifshitz equation
- Ishimori equation

See also the
list of non-linear partial differential equations

## Bibliography

## External links

- A collection of non-linear models and demo applets (in Monash University's Virtual Lab)
- Command and Control Research Program (CCRP)
- New England Complex Systems Institute: Concepts in Complex Systems
- Nonlinear Dynamics I: Chaos at MIT's OpenCourseWare
- Nonlinear Models Nonlinear Model Database of Physical Systems (MATLAB)
- The Center for Nonlinear Studies at Los Alamos National Laboratory
- FyDiK Software for simulations of nonlinear dynamical systems

nonlinearity in Bengali: অরৈখিকতা

nonlinearity in German: Nichtlineares
System

nonlinearity in Spanish: No linealidad

nonlinearity in Finnish: Epälineaarinen

nonlinearity in French: Non-linéarité

nonlinearity in Galician: Nonlinearidade

nonlinearity in Hebrew: מערכת לא לינארית

nonlinearity in Indonesian: Sistem
nonlinier

nonlinearity in Italian: Sistema non
lineare

nonlinearity in Japanese: 非線形システム論

nonlinearity in Korean: 비선형

nonlinearity in Dutch: Niet-lineair
systeem

nonlinearity in Norwegian: Ulineær

nonlinearity in Portuguese: Sistemas dinâmicos
não-lineares

nonlinearity in Russian: Нелинейная
система

nonlinearity in Swedish: Icke-linjär

nonlinearity in Chinese: 非線性