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Inmathematics andscience, anonlinear system (or anon-linear system) is asystem in which the change of the output is notproportional to the change of the input.[1][2] Nonlinear problems are of interest toengineers,biologists,[3][4][5]physicists,[6][7]mathematicians, and many otherscientists since most systems are inherently nonlinear in nature.[8] Nonlineardynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simplerlinear systems.
Typically, the behavior of a nonlinear system is described in mathematics by anonlinear system of equations, which is a set of simultaneousequations in which the unknowns (or the unknown functions in the case ofdifferential equations) appear as variables of apolynomial of degree higher than one or in the argument of afunction 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 alinear combination of the unknownvariables orfunctions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation islinear 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.
As nonlinear dynamical 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 assolitons,chaos,[9] andsingularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemblerandom behavior, it is in fact 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.
Some authors use the termnonlinear science for the study of nonlinear systems. This term 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.
Inmathematics, alinear map (orlinear function) is one which satisfies both of the following properties:
Additivity implies homogeneity for anyrationalα, and, forcontinuous functions, for anyrealα. For acomplexα, homogeneity does not follow from additivity. For example, anantilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle
An equation written as
is calledlinear if is a linear map (as defined above) andnonlinear otherwise. The equation is calledhomogeneous if and is ahomogeneous function.
The definition is very general in that can be any sensible mathematical object (number, vector, function, etc.), and the function can literally be anymapping, including integration or differentiation with associated constraints (such asboundary values). If containsdifferentiation with respect to, the result will be adifferential equation.
A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not alinear equation.
For a single equation of the form many methods have been designed; seeRoot-finding algorithm. In the case wheref is apolynomial, one has apolynomial equation such as The general root-finding algorithms apply to polynomial roots, but, generally they do not find all the roots, and when they fail to find a root, this does not imply that there is no roots. Specific methods for polynomials allow finding all roots or thereal roots; seereal-root isolation.
Solvingsystems of polynomial equations, that is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborate algorithms have been designed, such asGröbner base algorithms.[11]
For the general case of system of equations formed by equating to zero severaldifferentiable functions, the main method isNewton's method and its variants. Generally they may provide a solution, but do not provide any information on the number of solutions.
A nonlinearrecurrence relation defines successive terms of asequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are thelogistic map and the relations that define the variousHofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the relatednonlinear system identification and analysis procedures.[12] These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.
Asystem ofdifferential equations is said to be nonlinear if it is not asystem of linear equations. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are theNavier–Stokes equations in fluid dynamics and theLotka–Volterra equations in biology.
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 oflinearly independent solutions can be used to construct general solutions through thesuperposition principle. A good example of this is one-dimensional heat transport withDirichlet 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.
First orderordinary differential equations are often exactly solvable byseparation of variables, especially for autonomous equations. For example, the nonlinear equation
has as a general solution (and also the special solution corresponding to the limit of the general solution whenC tends to infinity). The equation is nonlinear because it may be written as
and the left-hand side of the equation is not a linear function of and its derivatives. Note that if the term were replaced with, the problem would be linear (theexponential decay problem).
Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yieldclosed-form solutions, though implicit solutions and solutions involvingnonelementary integrals are encountered.
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
The most common basic approach to studying nonlinearpartial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). Sometimes, the equation may be transformed into one or moreordinary differential equations, as seen inseparation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.
Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to usescale analysis to simplify a general, natural equation in a certain specificboundary value problem. For example, the (very) nonlinearNavier-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 thecharacteristics and using the methods outlined above for ordinary differential equations.
A classic, extensively studied nonlinear problem is the dynamics of a frictionlesspendulum under the influence ofgravity. UsingLagrangian mechanics, it may be shown[14] that the motion of a pendulum can be described by thedimensionless nonlinear equation
where gravity points "downwards" and is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use as anintegrating factor, which would eventually yield
which is an implicit solution involving anelliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in thenonelementary integral (nonelementary unless).
Another way to approach the problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest throughTaylor expansions. For example, the linearization at, called the small angle approximation, is
since for. This is asimple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at, corresponding to the pendulum being straight up:
since for. The solution to this problem involveshyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that 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, around which:
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.
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