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In mathematics and science, a **nonlinear system** is a system in which the change of the output is not proportional to the change of the input.^{ [1] }^{ [2] }^{ [3] } Nonlinear problems are of interest to engineers, biologists,^{ [4] }^{ [5] }^{ [6] } physicists,^{ [7] }^{ [8] } mathematicians, and many other scientists because most systems are inherently nonlinear in nature.^{ [9] } Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

**Science** is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

A **system** is a group of interacting or interrelated entities that form a unified whole. A system is delineated by its spatial and temporal boundaries, surrounded and influenced by its environment, described by its structure and purpose and expressed in its functioning. Systems are the subjects of study of systems theory.

- Definition
- Nonlinear algebraic equations
- Nonlinear recurrence relations
- Nonlinear differential equations
- Ordinary differential equations
- Partial differential equations
- Pendula
- Types of nonlinear dynamic behaviors
- Examples of nonlinear equations
- See also
- References
- Further reading
- External links

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.

In mathematics, an **equation** is a statement that asserts the equality of two expressions. The word *equation* and its cognates in other languages may have subtly different meanings; for example, in French an *équation* is defined as containing one or more variables, while in English any equality is an equation.

A **differential equation** is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In mathematics, a **polynomial** is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, *x*, is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

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,^{ [10] } 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.

In mathematics, **linearization** is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology.

In mathematics and physics, a **soliton** or **solitary wave** is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.

**Chaos theory** is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, meaning there is sensitive dependence on initial conditions. A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a hurricane in Texas.

Some authors use the term **nonlinear science** 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.

In logic,

negation, also called thelogical complement, is an operation that takes a proposition to another proposition "not ", written , which is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takestruthtofalsityand vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition is the proposition whose proofs are the refutations of .

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

In mathematics, a **linear map** is a mapping *V* → *W* between two modules that preserves the operations of addition and scalar multiplication.

- Additivity or superposition principle:
- Homogeneity:

The **superposition principle**, also known as **superposition property**, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input *A* produces response *X* and input *B* produces response *Y* then input produces response.

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

An equation written as

is called **linear** if is a linear map (as defined above) and **nonlinear** otherwise. The equation is called *homogeneous* if .

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

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

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.^{ [12] }

A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures.^{ [13] } These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.

A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.

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. 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. 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.

First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation

has as a general solution (and also *u* = 0 as a particular solution, corresponding to the limit of the general solution when *C* tends to infinity). The equation is nonlinear because it may be written as

and the left-hand side of the equation is not a linear function of *u* and its derivatives. Note that if the *u*^{2} term were replaced with *u*, the problem would be linear (the exponential decay problem).

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.

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

- Examination of any conserved quantities, especially in Hamiltonian systems
- Examination of dissipative quantities (see Lyapunov function) analogous to conserved quantities
- Linearization via Taylor expansion
- Change of variables into something easier to study
- Bifurcation theory
- Perturbation methods (can be applied to algebraic equations too)

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). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable.

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. 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.

Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations.

A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown^{ [14] } that the motion of a pendulum can be described by the dimensionless nonlinear equation

where gravity points "downwards" and is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use as an integrating factor, which would eventually yield

which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless ).

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. For example, the linearization at , called the small angle approximation, is

since for . This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at , corresponding to the pendulum being straight up:

since for . The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.

One more interesting linearization is possible around , around which :

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.

- Amplitude death – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system
- Chaos – values of a system cannot be predicted indefinitely far into the future, and fluctuations are aperiodic
- Multistability – the presence of two or more stable states
- Solitons – self-reinforcing solitary waves

- Algebraic Riccati equation
- Ball and beam system
- Bellman equation for optimal policy
- Boltzmann equation
- Colebrook equation
- General relativity
- Ginzburg–Landau theory
- Ishimori equation
- Kadomtsev–Petviashvili equation
- Korteweg–de Vries equation
- Landau–Lifshitz–Gilbert equation
- Liénard equation
- Navier–Stokes equations of fluid dynamics
- Nonlinear optics
- Nonlinear Schrödinger equation
- Power-flow study
- Richards equation for unsaturated water flow
- Self-balancing unicycle
- Sine-Gordon equation
- Van der Pol oscillator
- Vlasov equation

In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In mechanics and physics, **simple harmonic motion** is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

In physics, **equations of motion** are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum components and time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

In mathematics, a **recurrence relation** is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given; each further term of the sequence or array is defined as a function of the preceding terms.

In mathematics, the **Laplace operator** or **Laplacian** is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇^{2}. The Laplacian ∇·∇*f*(*p*) of a function *f* at a point *p*, is the rate at which the average value of *f* over spheres centered at *p* deviates from *f*(*p*) as the radius of the sphere shrinks towards 0. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

In mathematics, a **Green's function** of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions is its impulse response.

In mathematics, a **linear differential equation** is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

In mathematics, the **associated Legendre polynomials** are the canonical solutions of the **general Legendre equation**

In mathematics, a **linear approximation** is an approximation of a general function using a linear function. They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.

In mathematics and physics, the **Helmholtz equation**, named for Hermann von Helmholtz, is the linear partial differential equation

In mathematics, a **change of variables** is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.

A **parametric oscillator** is a driven harmonic oscillator in which the oscillations are driven by varying some parameter of the system at some frequency, typically different from the natural frequency of the oscillator. A simple example of a parametric oscillator is a child pumping a playground swing by periodically standing and squatting to increase the size of the swing's oscillations. The child's motions vary the moment of inertia of the swing as a pendulum. The "pump" motions of the child must be at twice the frequency of the swing's oscillations. Examples of parameters that may be varied are the oscillator's resonance frequency and damping .

A **pendulum** is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.

In mathematics, an **integro-differential equation** is an equation that involves both integrals and derivatives of a function.

A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the **vibrations of a circular membrane** of uniform thickness, attached to a rigid frame. Due to the phenomenon of resonance, at certain vibration frequencies, its resonant frequencies, the membrane can store vibrational energy, the surface moving in a characteristic pattern of standing waves. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental mode.

Miniaturizing components has always been a primary goal in the semiconductor industry because it cuts production cost and lets companies build smaller computers and other devices. Miniaturization, however, has increased dissipated power per unit area and made it a key limiting factor in integrated circuit performance. Temperature increase becomes relevant for relatively small-cross-sections wires, where it may affect normal semiconductor behavior. Besides, since the generation of heat is proportional to the frequency of operation for switching circuits, fast computers have larger heat generation than slow ones, an undesired effect for chips manufacturers. This article summaries physical concepts that describe the generation and conduction of heat in an integrated circuit, and presents numerical methods that model heat transfer from a macroscopic point of view.

For a plane curve *C* and a given fixed point *O*, the **pedal equation** of the curve is a relation between *r* and *p* where *r* is the distance from *O* to a point on *C* and *p* is the perpendicular distance from *O* to the tangent line to *C* at the point. The point *O* is called the *pedal point* and the values *r* and *p* are sometimes called the *pedal coordinates* of a point relative to the curve and the pedal point. It is also useful to measure the distance of *O* to the normal even though it is not an independent quantity and it relates to as .

- ↑ Boeing, G. (2016). "Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction".
*Systems*.**4**(4): 37. arXiv: 1608.04416 . doi:10.3390/systems4040037. - ↑ "Explained: Linear and nonlinear systems".
*MIT News*. Retrieved 2018-06-30. - ↑ "Nonlinear systems, Applied Mathematics - University of Birmingham".
*www.birmingham.ac.uk*. Retrieved 2018-06-30. - ↑ "Nonlinear Biology",
*The Nonlinear Universe*, The Frontiers Collection, Springer Berlin Heidelberg, 2007, pp. 181–276, doi:10.1007/978-3-540-34153-6_7, ISBN 9783540341529 - ↑ Korenberg, Michael J.; Hunter, Ian W. (March 1996). "The identification of nonlinear biological systems: Volterra kernel approaches".
*Annals of Biomedical Engineering*.**24**(2): 250–268. doi:10.1007/bf02667354. ISSN 0090-6964. - ↑ Mosconi, Francesco; Julou, Thomas; Desprat, Nicolas; Sinha, Deepak Kumar; Allemand, Jean-François; Vincent Croquette; Bensimon, David (2008). "Some nonlinear challenges in biology".
*Nonlinearity*.**21**(8): T131. Bibcode:2008Nonli..21..131M. doi:10.1088/0951-7715/21/8/T03. ISSN 0951-7715. - ↑ Gintautas, V. (2008). "Resonant forcing of nonlinear systems of differential equations".
*Chaos*.**18**(3): 033118. arXiv: 0803.2252 . Bibcode:2008Chaos..18c3118G. doi:10.1063/1.2964200. PMID 19045456. - ↑ Stephenson, C.; et., al. (2017). "Topological properties of a self-assembled electrical network via ab initio calculation".
*Sci. Rep*.**7**: 41621. Bibcode:2017NatSR...741621S. doi:10.1038/srep41621. PMC 5290745 . PMID 28155863. - ↑ de Canete, Javier, Cipriano Galindo, and Inmaculada Garcia-Moral (2011).
*System Engineering and Automation: An Interactive Educational Approach*. Berlin: Springer. p. 46. ISBN 978-3642202292 . Retrieved 20 January 2018. - ↑ Nonlinear Dynamics I: Chaos Archived 2008-02-12 at the Wayback Machine at MIT's OpenCourseWare
- ↑ Campbell, David K. (25 November 2004). "Nonlinear physics: Fresh breather".
*Nature*.**432**(7016): 455–456. Bibcode:2004Natur.432..455C. doi:10.1038/432455a. ISSN 0028-0836. PMID 15565139. - ↑ Lazard, D. (2009). "Thirty years of Polynomial System Solving, and now?".
*Journal of Symbolic Computation*.**44**(3): 222–231. doi:10.1016/j.jsc.2008.03.004. - ↑ Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013
- ↑ David Tong: Lectures on Classical Dynamics

- Diederich Hinrichsen and Anthony J. Pritchard (2005).
*Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness*. Springer Verlag. ISBN 9783540441250. - Jordan, D. W.; Smith, P. (2007).
*Nonlinear Ordinary Differential Equations*(fourth ed.). Oxford University Press. ISBN 978-0-19-920824-1. - Khalil, Hassan K. (2001).
*Nonlinear Systems*. Prentice Hall. ISBN 978-0-13-067389-3. - Kreyszig, Erwin (1998).
*Advanced Engineering Mathematics*. Wiley. ISBN 978-0-471-15496-9. - Sontag, Eduardo (1998).
*Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition*. Springer. ISBN 978-0-387-98489-6.

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