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In Mathematics , a nonlinear system is one whose behavior can't be expressed as a sum of the behaviors of its parts (or of their multiples.) In technical terms, the behavior of nonlinear systems is not subject to the principle of Superposition . Linear Systems are subject to superposition.

When a system is linear, people examining it can make certain mathematical assumptions and approximations about its behavior, allowing for simple computation of results. For instance, the height of a column of water poured into a glass is a simple function of the volume of water poured in, along with the diameter of the glass, making it easy to calculate the height of various possible volumes of water.

In nonlinear systems these assumptions cannot be made. Since nonlinear systems are not equal to the sum of their parts, they are often difficult (or impossible) to model, and their behavior with respect to a given variable (for example, time) is extremely difficult to predict. When modeling non-linear systems, therefore, it is common to approximate them as linear, where possible. The weather is famously non-linear, where simple changes in one part of the system produce complex effects throughout.

Some nonlinear systems are Exactly Solvable or Integrable , while others are known to be Chaotic , and thus have no simple or Closed Form solution. A possible example is that of Freak Wave s. Whilst some nonlinear systems and equations of general interest have been extensively studied, the general theory is poorly understood.


BACKGROUND


Linear systems

In Mathematics , a Linear Function \displaystyle f(x) is one which satisfies both of the following properties:

#Additivity: f(x + y) = f(x) + f(y) \
#Homogeneity: f(\alpha\,x) = \alpha\,f(x) \

These two rules, taken together, are often referred to as the principle of superposition. (It turns out that homogeneity follows from the additivity property in all cases where α is rational. In that case if the linear function is continuous, homogeneity is not an additional axiom to establish if the additivity property is established.) Important examples of linear operators include the Derivative considered as a Differential Operator , and many other operators 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.

Examples of linear operators
are matrices or linear combinations of powers of Partial Derivative s e.g.
:L=d_x^2 + d_y^2, where ''x'' and ''y'' are real variables.

A map ''F''(''u'') is a generalization of a Linear Operator . Equations involving maps include Linear Equation s 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 maps are
  • F(x) = x^2 \,, 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, u^2) \,, where ''u'', ''v'' are functions or numbers.



Nonlinear systems

Nonlinear equations and functions are of interest to Physicist s and Mathematician s because most physical systems are inherently nonlinear in nature. Physical examples of linear systems are relatively rare. Nonlinear equations are difficult to solve and give rise to interesting phenomena such as Chaos . A linear equation can be described by using a Linear Operator , \displaystyle L. A linear equation in some unknown \displaystyle u has the form:
:\displaystyle Lu=0

In order to solve any equation, one needs to decide in what Mathematical Space the solution \displaystyle u is found. It might be that \displaystyle u is a real number, a vector or perhaps a function with some properties. The solutions for linear equations can in general be described as a superposition of other solutions for the same equation. This makes linear equations particularly easy to solve.

Nonlinear equations are more complex. They are much harder to understand because of the lack of simple superposed solutions. For nonlinear equations, solutions generally do not form a Vector Space and commonly cannot 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

:x^2 - 1 = 0\,

and other polynomial equations.
Systems of nonlinear polynomial equation, however, are more complex.
Similarly,
first order nonlinear Ordinary Differential Equation such as

: rac{d u}{d x} = u^2

are easily solved (in this case, by Separation Of Variables ).

Higher order differential equations like

: rac{d^2 u}{d x^2} u + g(u) = 0\, , where \displaystyle g is any nonlinear function,

can be much more challenging, if exactly solvable at all.

For Partial Differential Equation s 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. The most common tactic for solving a nonlinear partial differential equation is to either transform it into a linear one or transform it into a (likely nonlinear) Ordinary Differential Equation , using for example the Similarity Transformation .

The differential equation of motion of a Simple Pendulum is non-linear:

:{d^2 heta\over dt^2}+{g\over \ell} \sin heta=0 \quad\quad\quad

Typically this is linearized by assuming small values of \displaystyle heta so that \displaystyle \sin heta \approx heta, so that

:{d^2 heta\over dt^2}+{g\over \ell} heta=0 \quad\quad\quad

For large values of heta, or if the non-linear behavior of the pendulum is of interest, the non-linear equation may be analyzed by Phase Plane methods, or else through the use of Elliptic Integrals .


TOOLS FOR SOLVING CERTAIN NON-LINEAR SYSTEMS

Today there are several tools for analyzing nonlinear equations. A few examples of these tools include: Implicit Function Theorem , contraction mapping principle and Bifurcation Theory . Perturbation techniques can be used to find approximate solutions to non-linear differential equations.


EXAMPLES OF NONLINEAR EQUATIONS




SEE ALSO



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