Nonlinear system
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/* Examples of nonlinear equations */ ball beam
:''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 system|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 [[homogenous|nonhomogenous]] system, which is linear apart from the presence of a function of the [[independent variable]]s, 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 [[physicist]]s and [[mathematician]]s 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 theory|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 (mathematics)|function]] (or map) <math>f(x)</math> is one which satisfies both of the following properties:
#Additivity: <math>f(x + y) = f(x) + f(y)\,</math>
#Homogeneity: <math>f(\alpha x) = \alpha f(x)\,</math>
An equation written as
:<math>f(x) = C\,</math>
is called '''linear''' if <math>f(x)</math> is [[linear map|linear]] (as defined above) and '''nonlinear''' otherwise. Note that <math>x</math> does not need to be a scalar (can be a [[vector]], [[function (mathematics)|function]], etc), and that <math>C</math> must not depend on <math>x</math>. The equation is called ''homogeneous'' if <math>C = 0</math>.
==Nonlinear algebraic equations==
Generally, nonlinear [[algebra]]ic problems are often exactly solvable, and if not they usually can be thoroughly understood through [[qualitative analysis|qualitative]] and [[numerical analysis|numeric]] analysis. As an example, the equation
:<math>x^2 + x - 1 = 0\,</math>
may be written as
:<math>f(x) = C \quad \mbox{where} \quad f(x) = x^2 + x \quad \mbox{and} \quad C = 1\,</math>
and is nonlinear because <math>f(x)</math> satisfies neither [[additivity]] nor [[homogeneity]] (the nonlinearity is due to the <math>x^2</math>). 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
:<math>x^5 - x - 1 = 0\,</math>
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 [[root (mathematics)|root]]s of <math>f(x) - C = 0</math>.
==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 sequence]]s.
==Nonlinear differential equations==
Problems involving nonlinear [[differential equation]]s 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 equation]]s are often exactly solvable by [[separation of variables]], especially for autonomous equations. For example, the nonlinear equation
:<math>\frac{d u}{d x} = -u^2\,</math>
will easily yield <math>u = (x + C)^{-1}</math> as a general solution which happens to be simpler than the solution to the linear equation <math>du/dx = -u</math>. The equation is nonlinear because it may be written as
:<math>\frac{d u}{d x} + u^2=0\,</math>
and the left-hand side of the equation is not a linear function of ''u'' and its derivatives. Note that if the ''u''² 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 integral]]s are encountered.
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
* Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s.
* 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 theory|Perturbation]] methods (can be applied to algebraic equations too).
===Partial differential equations===
The most common basic approach to studying nonlinear [[partial differential equation]]s 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 equation]]s, as seen in the [[Similar matrix | 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 (mathematics)|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 [[Method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations.
===Example: pendulum===
{{main|Pendulum (mathematics)}}
[[Image:PendulumLayout.png|thumb|Illustration of a pendulum.|right|200px]]
[[Image:PendulumLinearizations.png|thumb|Linearizations of a pendulum.|right|200px]]
A classic, extensively studied nonlinear problem is the dynamics of a [[Pendulum (mathematics)|pendulum]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation
:<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0\,</math>
where gravity is "down" and <math>\theta</math> is as shown in the figure at right. One approach to "solving" this equation is to use <math>\scriptstyle \frac{d \theta}{d t}</math> as an [[integrating factor]], which would eventually yield
:<math>\int \frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}} = t + C_1\,</math>
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 <math>C_0 = 0</math>).
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 expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is
:<math>\frac{d^2 \theta}{d t^2} + \theta = 0\,</math>
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a [[simple harmonic oscillator]] corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:
:<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0\,</math>
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> 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 <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:
:<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math>
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 portrait]]s 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==
<div style="-moz-column-count:2; column-count:2;">
* [[AC power flow model]]
* [[Ball and beam system]]
* [[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]]
</div>
See also the [[list of non-linear partial differential equations]]
==See also==
* [[Aleksandr Mikhailovich Lyapunov]]
* [[Dynamical system]]
==Bibliography==
*{{cite book
| last = Kreyszig
| first = Erwin
| authorlink = Erwin Kreyszig
| year = 1998
| title = Advanced Engineering Mathematics
| publisher = Wiley
| id = ISBN 0-471-15496-2
}}
*{{cite book
| last = Khalil
| first = Hassan K.
| year = 2001
| title = Nonlinear Systems
| publisher = Prentice Hall
| id = ISBN 0-13-067389-7
}}
* {{cite book
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard
| year= 2005
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness
| publisher= Springer Verlag
| id=ISBN 0-978-3-540-441250
}}
*{{cite book
| last = Sontag
| first = Eduardo
| authorlink = Eduardo D. Sontag
| year = 1998
| title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
| publisher = Springer
| id = ISBN 0-387-984895
}}
{{Ref-list}}
==External links==
* [http://vlab.infotech.monash.edu.au/simulations/non-linear/ A collection of non-linear models and demo applets] (in Monash University's Virtual Lab)
* [http://www.dodccrp.org/ Command and Control Research Program (CCRP)]
* [http://necsi.org/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]
* [http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]
* [http://www.hedengren.net/research/models.htm Nonlinear Models] Nonlinear Model Database of Physical Systems (MATLAB)
* [http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]
* [http://fydik.kitnarf.cz/ FyDiK] Software for simulations of nonlinear dynamical systems
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[[Category:Non-linear systems| ]]
[[Category:Dynamical systems]]
[[Category:Fundamental physics concepts]]
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