Everything about Nonlinear totally explained
» 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 isn't
linear, for example a system which doesn't satisfy the
superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to be solved for can't 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)
is one which satisfies both of the following properties:
- Additivity:
- Homogeneity:
An equation written as
»
is called
linear if
is
linear (as defined above) and
nonlinear otherwise. Note that
doesn't need to be a scalar (can be a
vector,
function, etc), and that
must not depend on
. The equation is called
homogeneous if
.
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
»
may be written as
»
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 can't be predicted.
Multistability - alternating between two or more exclusive states.
Aperiodic oscillations - functions that don't repeat values after some period (otherwise known as chaotic oscillations or chaos).
Examples of nonlinear equations
See also the list of non-linear partial differential equationsFurther Information
Get more info on 'Nonlinear'.
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