Nelinearni sistem

U matematici i nauci, nelinearni sistem je sistem u kome promena izlaza nije proporcionalna promeni na ulazu. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. 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, 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.

Neki autori koriste termin nelinearna nauka for the study of nonlinear systems. This 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.

— Stanislaw Ulam

Definicija

U matematici, a linear map (or linear function) $f(x)$ is one which satisfies both of the following properties:

• Additivity or superposition principle: $\textstyle f(x+y)=f(x)+f(y);$
• Homogeneity: $\textstyle f(\alpha x)=\alpha f(x).$

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

$f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)$

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 homogeneous if $C=0$.

The definition $f(x)=C$ is very general in that $x$ can be any sensible mathematical object (number, vector, function, etc.), and the function $f(x)$ can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If $f(x)$ contains differentiation with respect to $x$, the result will be a differential equation.

Nelinearne algebrske jednačine

Glavni članci: Algebraic equation i System of polynomial equations

Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,

$x^{2}+x-1=0\,.$

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 complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.

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Spoljašnje veze

• Command and Control Research Program (CCRP)
• New England Complex Systems Institute: Concepts in Complex Systems
• Nonlinear Dynamics I: Chaos at MIT's OpenCourseWare
• Nonlinear Model Library – (in MATLAB) a Database of Physical Systems
• The Center for Nonlinear Studies at Los Alamos National Laboratory