In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input.[1] [2] Nonlinear problems are of interest to engineers, biologists,[3] [4] physicists,[5] [6] mathematicians, and many other scientists since most systems are inherently nonlinear in nature.[7] Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which 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 polynomial of degree one.In other words, in a nonlinear system 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. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear 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 as solitons, chaos,[8] and singularities 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 resemble random 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 term nonlinear science for the study of nonlinear systems. This term is disputed by others:
In mathematics, a linear map (or linear function)
f(x)
stylef(x+y)=f(x)+f(y);
stylef(\alphax)=\alphaf(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 example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle
f(\alphax+\betay)=\alphaf(x)+\betaf(y)
An equation written as
f(x)=C
is called linear if
f(x)
C=0
f(x)
The definition
f(x)=C
x
f(x)
f(x)
x
A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not a linear equation.
For a single equation of the form
f(x)=0,
x2+x-1=0.
Solving systems of polynomial equations, that is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborated algorithms have been designed, such as Gröbner base algorithms.[9]
For the general case of system of equations formed by equating to zero several differentiable functions, the main method is Newton's method and its variants. Generally they may provide a solution, but do not provide any information on the number of solutions.
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 discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures.[10] These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.
A system of differential equations is said to be nonlinear if it is not a system 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 the Navier–Stokes equations in fluid dynamics and the Lotka–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 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.
First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation
| du | |
| dx |
=-u2
has
| u= | 1 |
| x+C |
u=0,
| du | |
| dx |
+u2=0
and the left-hand side of the equation is not a linear function of
u
u2
u
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:
See main article: Nonlinear partial differential equation.
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 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 ordinary differential equation(s) is solvable.
Another common (though less mathematical) tactic, often exploited 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.
See main article: Pendulum (mathematics).
A classic, extensively studied nonlinear problem is the dynamics of a frictionless pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown[12] that the motion of a pendulum can be described by the dimensionless nonlinear equation
| d2\theta | |
| dt2 |
+\sin(\theta)=0
where gravity points "downwards" and
\theta
d\theta/dt
| \int{ | d\theta |
| \sqrt{C0+2\cos(\theta) |
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 unless
C0=2
Another way to approach the problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at
\theta=0
| d2\theta | |
| dt2 |
+\theta=0
since
\sin(\theta) ≈ \theta
\theta ≈ 0
\theta=\pi
| d2\theta | |
| dt2 |
+\pi-\theta=0
since
\sin(\theta) ≈ \pi-\theta
\theta ≈ \pi
|\theta|
One more interesting linearization is possible around
\theta=\pi/2
\sin(\theta) ≈ 1
| d2\theta | |
| 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.