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Nonlinearity

Nonlinear systems are mathematically represented systems in which the behavior of the system is not expressible as a linear function of its descriptors. In such a system, features such as superposition exist and the equation can often be split into "sum of its parts" which means certain kinds of assumption, approximation and mathematical approaches are possible.

By contrast, a nonlinear equation or system is not subject to such limits and usually cannot be simplified this way. It may exhibit behaviour and results which are extremely hard (or impossible) to calculate or predict under current knowledge or technology, chaos effects, strange attractors, and freak effects. Whilst some nonlinear systems and equations of general interest have been extensively studied, the vast majority are poorly understood if at all.

Nonlinear systems are probably easiest understood as "everything except the relatively few systems which prove to be linear".

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Background

Linear systems

In mathematics, a linear function f(x) is one which satisfies the following two properties:

• Additivity: f(x + y) = f(x) + f(y)
• Homogeneity: fx) = αf(x) for all α

Systems that satisfy both additivity and homogeneity are considered to be linear systems. These two rules, taken together, are often referred to as the principle of superposition. Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. When an equation can be expressed in linear form, it becomes particularly easy to solve because it can be broken down into smaller pieces that may be solved individually.

Nonlinear systems

Nonlinear equations and functions are of interest to physicists and mathematicians because they are hard to solve and give rise to interesting phenomena such as chaos. A linear equation can be described by a using a linear operator, L. A linear equation in some unknown u has the form

Lu = 0.

Examples of linear operators are matrices or linear combinations of powers of partial derivatives e.g.

$L=d_x^2 + d_y$, where x and y are real variables.

A map F(u) is a generalization of a linear operator. Equations involving maps include linear equations, and nonlinear equations as well as nonlinear systems (the last is a misnomer stemming from matrix equation 'systems', a nonlinear equation can be a scalar valued or matrix valued equation). Examples of a maps are

• F(x) = x2, where x a real number;
• $F(u)=-d_x^2 u + g(u)$, where u is a function u(x) and x is a real number and g is a function;
• F(u,v) = (u + v,u2), where u, v are functions or numbers.

A nonlinear equation is an equation of the form F(u) = 0, for some unknown u.

In order to solve any equation, one needs to decide in what mathematical space the solution u is found. It might be that u is a real number, a vector or perhaps a function with some properties.

The solutions of linear equations can in general be described as a superposition of other solutions of the same equation. This makes linear equations particularly easy to solve.

Nonlinear equations are more complex, and much harder to understand because of their lack of simple superposed solutions. For nonlinear equations the solutions to the equations do not in general form a vector space and cannot (in general) be superposed (added together) to produce new solutions. This makes solving the equations much harder than in linear systems.

Specific nonlinear equations

Some nonlinear equations are well understood, for example

x2 - 1 = 0

and other polynomial equations. Systems of nonlinear polynomial equations, however, are more complex. Similarly, first order nonlinear ordinary differential equation such as

dxu = u2

are easily solved (in this case, by separation of variables). Higher order differential equations like

$d_x^2 u + g(u)=0$ , where g is any nonlinear function,

can be much more challenging. For partial differential equations the picture is even poorer, although a number of results involving existence of solutions, stability of a solution and dynamics of solutions have been proven.

Tools for solving certain non-linear systems

Today there are several tools for analyzing nonlinear equations, to mention a few: Implicit function theorem, Contraction mapping principle and the theory of bifurcations.

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