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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ...
A linear system is a model of a system based on some kind of linear operator. ...
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 nonlinear systems, therefore, it is common to approximate them as linear, where possible. The weather is famously nonlinear, 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 waves. Whilst some nonlinear systems and equations of general interest have been extensively studied, the general theory is poorly understood. In theoretical physics, an exactly solvable model or integrable model refers to a physical model, a physical theory, or set of differential equations whose exact solution may be calculated analytically in terms of elementary or special functions; the adjective integrable therefore implies solvablility. ...
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ...
A plot of the Lorenz attractor for values r = 28, Ïƒ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under specific conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). ...
In mathematics, closed form can mean: a finitary expression, rather than one involving (for example) an infinite series, or use of recursion  this meaning usually occurs in a phrase like solution in closed form and one also says closed formula; a closed differential form: see Closed and exact differential forms. ...
The Draupner wave, a single giant wave measured on New Years Day 1995, finally confirmed the existence of freak waves, which had previously been considered nearmythical Freak waves, also known as rogue waves or monster waves, are relatively large and spontaneous ocean surface waves which can sink even...
Background
Linear systems In mathematics, a linear function is one which satisfies both of the following properties: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
The word linear comes from the Latin word linearis, which means created by lines. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
 Additivity:
 Homogeneity:
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. For a nontechnical overview of the subject, see Calculus. ...
In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...
In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ...
In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...
Examples of linear operators are matrices or linear combinations of powers of partial derivatives e.g. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...
 , 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 maps are In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. ...

 , where x a real number;
 , where u is a function u(x) and x is a real number and g is a function;
 , where u, v are functions or numbers.
Nonlinear systems Nonlinear equations and functions are of interest to physicists and mathematicians 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, . A linear equation in some unknown has the form: Articles with similar titles include physician, a person who practices medicine. ...
Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ...
A plot of the Lorenz attractor for values r = 28, Ïƒ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under specific conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In order to solve any equation, one needs to decide in what mathematical space the solution is found. It might be that 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. Space has been an interest for philosophers and scientists for much of human history. ...
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. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational firstorder reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
Specific nonlinear equations Some nonlinear equations are well understood, for example and other polynomial equations. Systems of nonlinear polynomial equation, however, are more complex. Similarly, first order nonlinear ordinary differential equation such as 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 derivatives with respect to that variable. ...
are easily solved (in this case, by separation of variables). Higher order differential equations like In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. ...
 , where 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. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
The differential equation of motion of a simple pendulum is nonlinear: A gravity pendulum is a weight on the end of a rigid rod, which, when given some initial lift from the vertical position, will swing back and forth under the influence of gravity over its central (lowest) point. ...
Typically this is linearized by assuming small values of so that , so that For large values of θ, or if the nonlinear behavior of the pendulum is of interest, the nonlinear equation may be analyzed by phase plane methods, or else through the use of elliptic integrals. Systems of differential equations are collectively of the general form dx/dt = Cx where C may be any combination of constants in order to create linear combinations with x on the right side; here x is in bold to indicate it is actually a vector, not a scalar. ...
In integral calculus, an elliptic integral is any function f which can be expressed in the form where R is a rational function of its two arguments, P is the square root of a polynomial of degree 3 or 4 (a cubic or quartic) with no repeated roots, and c...
Tools for solving certain nonlinear 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 nonlinear differential equations. In multivariable calculus, a branch of mathematics, the implicit function theorem is a tool which allows relations to be converted to functions. ...
In mathematics, specifically in the study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden qualitative or topological change in the systems longterm dynamical behaviour. ...
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ...
Examples of nonlinear equations AC power flow model is a model used in electrical engineering to analyze power grids. ...
An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...
The NavierStokes equations, named after ClaudeLouis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ...
Fluid dynamics is the subdiscipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ...
In mathematics and physics, a soliton is a selfreinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed; solitons are caused by a delicate balance between nonlinear and dispersive effects in the medium. ...
Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. ...
Weather is a term that encompasses phenomena in the atmosphere of a planet. ...
The problem of creating a robotic unicycle, a selfpowered unicycle that balances itself in three dimensions, is an interesting problem in robotics and control theory. ...
The Kortewegde Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ...
The SineGordon equation is a partial differential equation for a function of two real variables, x and t, given as follows: The name is a pun on the KleinGordon equation. ...
In theoretical physics, the nonlinear SchrÃ¶dinger equation is a nonlinear version of SchrÃ¶dingers equation in two dimensions. ...
In physics, GinzburgLandau theory is a mathematical theory used to model superconductivity. ...
Bellman equations occur in dynamic programming. ...
The Richards equation represents the movement of water in unsaturated soils, and was formulated by Lorenzo A. Richards in 1931. ...
See also Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6, 1857  November 3, 1918, all new style) was a Russian mathematician, mechanician and physicist. ...
The Lorenz attractor is an example of a nonlinear dynamical system. ...
Bibliography  Kreyszig, Erwin (1998). Advanced Engineering Mathematics. Wiley. ISBN 0471154962.
 Khalil, Hassan K. (2001). Nonlinear Systems. Prentice Hall. ISBN 0130673897.
 Diederich Hinrichsen and Anthony J. Pritchard (2005). Mathematical Systems Theory I  Modelling, State Space Analysis, Stability and Robustness. Springer Verlag. ISBN 09783540441250.
 Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition. Springer. ISBN 0387984895.
Erwin Kreyszig is a Professor of Mathematics at Ohio State University. ...
// Diederich Hinrichsen (born 17. ...
Eduardo Sontag (b. ...
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