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This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity (disambiguation).

In mathematics, a nonlinear system is a system which is not linear, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A linear system is a mathematical model of a System based on the use of a Linear operator. e. a system which does not satisfy the superposition principle. In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear sum of independent components. A nonhomogenous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known. Dependent variables and independent variables refer to values that change in relationship to each other

Generally, nonlinear problems are difficult (if possible) to solve and are much less understandable than linear problems. Even if not exactly solvable, the outcome of a linear problem is rather predictable, while the outcome of a nonlinear is inherently not.

Nonlinear problems are of interest to physicists and mathematicians because most physical systems are inherently nonlinear in nature. A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning A mathematician is a person whose primary area of study and research is the field of Mathematics. Physical examples of linear systems are not very common. Nonlinear equations are difficult to solve and give rise to interesting phenomena such as chaos. In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that The weather is famously nonlinear, where simple changes in one part of the system produce complex effects throughout.

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Definition

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

  1. Additivity: f(x + y) = f(x) + f(y)\,
  2. Homogeneity: f(\alpha x) = \alpha f(x)\,

An equation written as

f(x) = C\,

is called linear if f(x) is linear (as defined above) and nonlinear otherwise. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The word linear comes from the Latin word linearis, which means created by lines. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Note that x does not need to be a scalar (can be a vector, function, etc), and that C must not depend on x. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The equation is called homogeneous if C = 0.

Nonlinear algebraic equations

Generally, nonlinear algebraic problems are often exactly solvable, and if not they usually can be thoroughly understood through qualitative and numeric analysis. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Qualitative research is a field of inquiry that crosscuts disciplines and subject matters. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) As an example, the equation

x^2 + x - 1 = 0\,

may be written as

f(x) = C \quad \mbox{where} \quad f(x) = x^2 + x \quad \mbox{and} \quad C = 1\,

and is nonlinear because f(x) satisfies neither additivity nor homogeneity (the nonlinearity is due to the x2). Though nonlinear, this simple example may be solved exactly (via the quadratic formula) and is very well understood. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. On the other hand, the nonlinear equation

x^5 - x - 1 = 0\,

is not exactly solvable (see quintic equation), though it may be qualitatively analyzed and is well understood, for example through making a graph and examining the roots of f(x) − C = 0. In Mathematics, a quintic equation is a Polynomial Equation of degree five This article is about the zeros of a function which should not be confused with the value at zero.

Nonlinear recurrence relations

A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. "Difference equation" redirects here It should not be confused with a Differential equation. In Mathematics, a sequence is an ordered list of objects (or events Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. The logistic map is a Polynomial mapping of degree 2, often cited as an archetypal example of how complex chaotic behaviour can arise from very simple In Mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear Recurrence relations.

Nonlinear differential equations

Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are very problem dependent. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the

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. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems 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. In Mathematics, the Dirichlet (or first type) boundary condition is a type of Boundary condition, named after Johann Peter Gustav Lejeune 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. In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems

Ordinary differential equations

First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an For example, the nonlinear equation

\frac{d u}{d x} = -u^2\,

will easily yield u = (x + C) − 1 as a general solution which happens to be simpler than the solution to the linear equation du / dx = − u. The equation is nonlinear because it may be written as

\frac{d u}{d x} + u^2=0\,

and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u² term were replaced with u, the problem would be linear (the exponential decay problem). A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value

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. In Mathematics, a nonelementary integral is an Integral for which it can be shown that there exists no formula in terms of Elementary functions ' (i

Common methods for the qualitative analysis of nonlinear ordinary differential equations include:

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 even linear). In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in the similarity transform or separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In Linear algebra, two n -by- n matrices A and B over the field K are called similar if there exists In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an

Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. Scale analysis is a powerful tool used in the mathematical sciences for the simplification of Equations with many terms In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints 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. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such

Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations. In Mathematics, the method of characteristics is a technique for solving Partial differential equations.

Example: pendulum

Main article: Pendulum (mathematics)
Illustration of a pendulum.
Illustration of a pendulum. The mathematics of Pendulums are in general quite complicated
Linearizations of a pendulum.
Linearizations of a pendulum.

A classic, extensively studied nonlinear problem is the dynamics of a pendulum. The mathematics of Pendulums are in general quite complicated Using Lagrangian mechanics, it may be shown[1] that the motion of a pendulum can be described by the dimensionless nonlinear equation

\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0\,

where gravity is "down" and θ is as shown in the figure at right. Lagrangian mechanics is a re-formulation of Classical mechanics that combines Conservation of momentum with Conservation of energy. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units One approach to "solving" this equation is to use \scriptstyle \frac{d \theta}{d t} as an integrating factor, which would eventually yield

\int \frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}} = t + C_1\,

which is an implicit solution involving an elliptic integral. In Mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given Ordinary differential equation. In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary even if C0 = 0). In Mathematics, a nonelementary integral is an Integral for which it can be shown that there exists no formula in terms of Elementary functions ' (i

Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives For example, the linearization at θ = 0, called the small angle approximation, is

\frac{d^2 \theta}{d t^2} + \theta = 0\,

since \sin(\theta) \approx \theta for \theta \approx 0. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. This article is about the harmonic oscillator in classical mechanics Another linearization would be at θ = π, corresponding to the pendulum being straight up:

\frac{d^2 \theta}{d t^2} + \pi - \theta = 0\,

since \sin(\theta) \approx \pi - \theta for \theta \approx \pi. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that | θ | will usually grow without limit, though bounded solutions are possible. In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.

One more interesting linearization is possible around θ = π / 2, around which \sin(\theta) \approx 1:

\frac{d^2 \theta}{d t^2} + 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. A phase portrait is a geometric representation of the trajectories of a Dynamical system in the Phase plane.

Metaphorical use

Engineers often use the term nonlinear to refer to irrational behavior, with the implication that the person who has become nonlinear is on the edge of losing control or even having a nervous breakdown. Mental breakdown (also known as nervous breakdown or snapping) is a non-medical term used to describe a sudden acute attack of Mental illness such as

Types of nonlinear behaviors

Examples of nonlinear equations

See also the list of non-linear partial differential equations

See also

Bibliography

  1. ^ David Tong: Lectures on Classical Dynamics

External links

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