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Nonlinear system

“Nonlinear dynamics” redirects here. For the journal,

see Nonlinear Dynamics (journal).

This article is about “nonlinearity” in mathematics,

physics and other sciences. For video and film editing,

see   Non-linear editing system. For other uses, see

nonlinearity (disambiguation).

In  physics   and other sciences, a  nonlinear system, in

contrast to a linear system, is a system which does not sat-

isfy the superposition principle – meaning that the output

of a nonlinear system is not directly proportional to theinput.

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 poly-

nomial of degree one. In other words, in a nonlinear sys-

tem 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 interms 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[1] 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.

1 Definition

In  mathematics, a linear function (or map)  f (x)   is one

which satisfies both of the following properties:

•   Additivity or  superposition:   f (x +  y) =  f (x) +f (y);

•   Homogeneity: f (αx) = αf (x).

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 exam-

ple, an antilinear map is additive but not homogeneous.

The conditions of additivity and homogeneity are often

combined in the superposition principle

f (αx + βy) = αf (x) + βf (y)

An equation written as

f (x) = C 

is called linear if f (x) is a linear map (as defined above)

and  nonlinear  otherwise. The equation is called homo-

 geneous  if C  = 0 .

The definition f (x) = C  is very general in that  x  can be

any sensible mathematical object (number, vector, func-

tion, etc.), and the function  f (x)   can literally be any

mapping, including integration or differentiation with as-

sociated constraints (such as boundary values). If  f (x)contains differentiation with respect to x  , the result will

be a  differential equation.

2 Nonlinear algebraic equations

Main article: Algebraic equation

Main article: Systems of polynomial equations

Nonlinear   algebraic equations, which are also called

 polynomial equations , are defined by equating

polynomials to zero. For example,

x2 + x − 1 = 0 .

1

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2   4 NONLINEAR DIFFERENTIAL EQUATIONS 

For a single polynomial equation, root-finding algorithms

can be used to find solutions to the equation (i.e., sets

of values for the variables that satisfy the equation).

However, systems of algebraic equations are more com-

plicated; their study is one motivation for the field of

algebraic geometry, a difficult branch of modern math-

ematics. It is even difficult to decide whether a given al-gebraic system has complex solutions (see Hilbert’s Null-

stellensatz). Nevertheless, in the case of the systems with

a finite number of complex solutions, these systems of

polynomial equations  are now well understood and effi-

cient methods exist for solving them.[2]

3 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 var-

ious  Hofstadter sequences. Nonlinear discrete models

that represent a wide class of nonlinear recurrence rela-

tionships include the NARMAX (Nonlinear Autoregres-

sive Moving Average with eXogenous inputs) model and

the related nonlinear system identification  and analysis

procedures.[3] These approaches can be used to study a

wide class of complex nonlinear behaviors in the time,

frequency, and spatio-temporal domains.

4 Nonlinear differential equations

A system of differential equations is said to be nonlinear

if it is not a linear system. Problems involving nonlinear

differential equations are extremely diverse, and methods

of solution or analysis are problem dependent. Examples

of nonlinear differential equations are the Navier–Stokes

equations in fluid dynamics and the Lotka–Volterra equa-tions in biology.

One of the greatest difficulties of nonlinear problems is

that it is not generally possible to combine known solu-

tions 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 so-

lution of which can be written as a time-dependent lin-

ear 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.

4.1 Ordinary differential equations

First order ordinary differential equations are often ex-

actly solvable by   separation of variables, especially for

autonomous equations. For example, the nonlinear equa-

tion

d u

d x  = −u2

has u   =   1

x+C  as a general solution (and also  u  = 0 as a

particular solution, corresponding to the limit of the gen-

eral solution when C  tends to the infinity). The equation

is nonlinear because it may be written as

d u

d x + u2 = 0

and the left-hand side of the equation is not a linear func-

tion 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 encoun-

tered.

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 (seeLyapunov

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).

4.2 Partial differential equations

See also:  List of nonlinear 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 separation of variables,which is always useful whether or not the resulting ordi-

nary differential equation(s) is solvable.

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4.3 Pendula   3

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 spe-

cific boundary value problem. For example, the (very)

nonlinear Navier-Stokes equations can be simplified into

one linear partial differential equation in the case of tran-

sient, laminar, one dimensional flow in a circular pipe; thescale analysis provides conditions under which the flow is

laminar and one dimensional and also yields the simpli-

fied equation.

Other methods include examining the characteristics and

using the methods outlined above for ordinary differential

equations.

4.3 Pendula

Main article: Pendulum (mathematics)

A classic, extensively studied nonlinear problem is the

gravityhinge

rigid massless rod

θ

mass

Illustration of a pendulum

Linearizations of a pendulum

dynamics of a   pendulum   under influence of   gravity.

Using   Lagrangian mechanics, it may be shown[4] that

the motion of a pendulum can be described by the

dimensionless nonlinear equation

d2θ

dt2  + sin(θ) = 0

where gravity points “downwards” and θ  is the angle the

pendulum forms with its rest position, as shown in the fig-

ure at right. One approach to “solving” this equation is to

use  dθ/dt  as an integrating factor, which would eventu-

ally yield

∫   dθ√ C 0 + 2 cos(θ)

= t + C 1

which is an implicit solution involving an elliptic integral.

This “solution” generally does not have many uses be-

cause 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. Forexample, the linearization atθ = 0 , called the smallangle

approximation, is

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4   8 SEE ALSO 

d2θ

dt2  + θ = 0

since sin(θ)   ≈   θ   for  θ   ≈   0   . This is a   simple har-

monic oscillator corresponding to oscillations of the pen-

dulum near the bottom of its path. Another linearization

would be at θ =  π  , corresponding to the pendulum being

straight up:

d2θ

dt2  + π − θ = 0

since sin(θ)  ≈   π −  θ  for θ  ≈   π   . The solution to this

problem involves hyperbolic sinusoids, and note that un-

like the small angle approximation, this approximation is

unstable, meaning that |θ| will usually grow without limit,

though bounded solutions are possible. This correspondsto the difficulty of balancing a pendulum upright, it is lit-

erally an unstable state.

One more interesting linearization is possible around θ =π/2 , around which sin(θ) ≈  1  :

d2θ

dt2 + 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.

5 Types of nonlinear behaviors

•   Classical chaos – the behavior of a system cannot be

predicted.

•   Multistability – alternating between two or more ex-

clusive states.

•   Aperiodic   oscillations – functions that do not re-peat values after some period (otherwise known as

chaotic oscillations or chaos).

•   Amplitude death  – any oscillations present in the

system cease due to some kind of interaction with

other system or feedback by the same system.

•   Solitons – self-reinforcing solitary waves

6 Examples of nonlinear equations

•  AC power flow model

•   Algebraic Riccati equation

•  Ball and beam system

•  Bellman equation for optimal policy

•  Boltzmann transport equation

•   Colebrook 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–Gilbert equation

•  Ishimori equation

•  Van der Pol equation

•  Liénard equation

•  Vlasov equation

See also the list of nonlinear partial differential equations

7 Software for solving nonlinearsystems

•   interalg – A solver from  OpenOpt /  FuncDesigner

frameworks for searching either any or all solutions

of nonlinear algebraic equations system

•   A collection of non-linear models and demo applets

(in Monash University’s Virtual Lab)

•   FyDiK – Software for simulations of nonlinear dy-

namical systems

8 See also

•   Aleksandr Mikhailovich Lyapunov

•   Dynamical system

•   Initial condition

•   Interaction

•  Linear system

•  Mode coupling

•   Vector soliton

•   Volterra series

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5

9 References

[1]   Nonlinear Dynamics I: Chaos at MIT’s OpenCourseWare

[2] Lazard, D. (2009). “Thirty years of Polynomial System

Solving, and now?".  Journal of Symbolic Computation  44

(3): 222–231.  doi:10.1016/j.jsc.2008.03.004.

[3] Billings S.A. “Nonlinear System Identification: NAR-

MAX Methods in the Time, Frequency, and Spatio-

Temporal Domains”. Wiley, 2013

[4]   David Tong: Lectures on Classical Dynamics

10 Further reading

•   Diederich Hinrichsen   and Anthony J. Pritchard

(2005).   Mathematical Systems Theory I - Mod-

elling, State Space Analysis, Stability and Robustness .Springer Verlag.  ISBN 9783540441250.

•   Jordan, D. W.; Smith, P. (2007).   Nonlinear Or-

dinary Differential Equations   (fourth ed.). Oxford

University Press. ISBN 978-0-19-920824-1.

•  Khalil, Hassan K. (2001).  Nonlinear Systems . Pren-

tice Hall. ISBN 0-13-067389-7.

•   Kreyszig, Erwin   (1998).   Advanced Engineering

Mathematics . Wiley.  ISBN 0-471-15496-2.

•   Sontag, Eduardo   (1998).   Mathematical Control 

Theory: Deterministic Finite Dimensional Systems.Second Edition. Springer.  ISBN 0-387-98489-5.

11 External links

•  Command and Control Research Program (CCRP)

•   New England Complex Systems Institute: Concepts

in Complex Systems

•   Nonlinear Dynamics I: Chaos   at   MIT’s Open-

CourseWare

•   Nonlinear Models  Nonlinear Model Database of

Physical Systems (MATLAB)

•   The Center for Nonlinear Studies at Los Alamos Na-

tional Laboratory

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6   12 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 

12 Text and image sources, contributors, and licenses

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