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Nonlinear system

Index Nonlinear system

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]

103 relations: Aleksandr Lyapunov, Algebraic equation, Algebraic geometry, Algebraic Riccati equation, Amplitude death, Antilinear map, Ball and beam, Bellman equation, Bifurcation theory, Biologist, Boltzmann equation, Boundary value problem, Chaos theory, Closed-form expression, Complex number, Conserved quantity, Continuous function, Darcy friction factor formulae, Derivative, Diederich Hinrichsen, Differential equation, Dimensionless quantity, Dirichlet boundary condition, Dynamical system, Elliptic integral, Engineer, Equation, Exponential decay, Fluid dynamics, Function (mathematics), General relativity, Ginzburg–Landau theory, Gravity, Hamiltonian system, Hilbert's Nullstellensatz, Hofstadter sequence, Hyperbolic function, Initial condition, Integrating factor, Interaction, Ishimori equation, Korteweg–de Vries equation, Lagrangian mechanics, Landau–Lifshitz–Gilbert equation, Liénard equation, Linear combination, Linear independence, Linear map, Linear system, Linearization, ..., Logistic map, Lotka–Volterra equations, Lyapunov function, Map (mathematics), Mathematician, Mathematics, MATLAB, Method of characteristics, Mode coupling, Multistability, Navier–Stokes equations, Negation, Nonelementary integral, Nonlinear optics, Nonlinear Schrödinger equation, Nonlinear system identification, Ordinary differential equation, Partial differential equation, Pendulum (mathematics), Periodic function, Perturbation theory, Phase portrait, Physicist, Polynomial, Power-flow study, Proportionality (mathematics), Randomness, Rational number, Real number, Recurrence relation, Richards equation, Root-finding algorithm, Scale analysis (mathematics), Science, Scientist, Self-balancing unicycle, Separation of variables, Sequence, Simple harmonic motion, Sine-Gordon equation, Singularity (mathematics), Soliton, Stanislaw Ulam, Superposition principle, System, System of equations, System of polynomial equations, Taylor series, Van der Pol oscillator, Variable (mathematics), Vector soliton, Vlasov equation, Volterra series. Expand index (53 more) »

Aleksandr Lyapunov

Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в,; – November 3, 1918) was a Russian mathematician, mechanician and physicist.

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Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.

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Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

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Algebraic Riccati equation

An algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time.

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Amplitude death

In the theory of dynamical systems, amplitude death is complete cessation of oscillations.

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Antilinear map

In mathematics, a mapping f:V\to W from a complex vector space to another is said to be antilinear (or conjugate-linear) if for all a, \, b \, \in \mathbb and all x, \, y \, \in V, where \bar and \bar are the complex conjugates of a and b respectively.

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Ball and beam

The ball and beam system consists of a long beam which can be tilted by a servo or electric motor together with a ball rolling back and forth on top of the beam.

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Bellman equation

A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming.

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Bifurcation theory

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.

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Biologist

A biologist, is a scientist who has specialized knowledge in the field of biology, the scientific study of life.

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Boltzmann equation

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.

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Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.

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Chaos theory

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

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Closed-form expression

In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Conserved quantity

In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables whose value remains constant along each trajectory of the system.

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Darcy friction factor formulae

In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the Darcy friction factor, a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open-channel flow.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Diederich Hinrichsen

Diederich Hinrichsen (born 17 February 1939) is a German mathematician who, together with Hans W. Knobloch, established the field of dynamical systems theory and control theory in Germany.

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Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

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Dimensionless quantity

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned.

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Dirichlet boundary condition

In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859).

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Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.

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Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

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Engineer

Engineers, as practitioners of engineering, are people who invent, design, analyze, build, and test machines, systems, structures and materials to fulfill objectives and requirements while considering the limitations imposed by practicality, regulation, safety, and cost.

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Equation

In mathematics, an equation is a statement of an equality containing one or more variables.

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Exponential decay

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value.

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Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

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Ginzburg–Landau theory

In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity.

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Gravity

Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.

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Hamiltonian system

A Hamiltonian system is a dynamical system governed by Hamilton's equations.

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Hilbert's Nullstellensatz

Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see Satz) is a theorem that establishes a fundamental relationship between geometry and algebra.

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Hofstadter sequence

In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations.

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Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

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Initial condition

In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t.

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Integrating factor

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials.

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Interaction

Interaction is a kind of action that occur as two or more objects have an effect upon one another.

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Ishimori equation

The Ishimori equation (IE) is a partial differential equation proposed by the Japanese mathematician.

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Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries equation (KdV equation for short) is a mathematical model of waves on shallow water surfaces.

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Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

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Landau–Lifshitz–Gilbert equation

In physics, the Landau–Lifshitz–Gilbert equation, named for Lev Landau and Evgeny Lifshitz and T. L. Gilbert, is a name used for a differential equation describing the precessional motion of magnetization in a solid.

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Liénard equation

In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Marie Liénard.

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Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Linear independence

In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Linear system

A linear system is a mathematical model of a system based on the use of a linear operator.

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Linearization

In mathematics, linearization is finding the linear approximation to a function at a given point.

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Logistic map

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations.

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Lotka–Volterra equations

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey.

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Lyapunov function

In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE.

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Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

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Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks.

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Method of characteristics

In mathematics, the method of characteristics is a technique for solving partial differential equations.

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Mode coupling

In the term mode coupling, as used in physics and electrical engineering, the word "mode" refers to eigenmodes of an idealized, "unperturbed", linear system.

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Multistability

In a dynamical system, multistability is the property of having multiple stable equilibrium points in the vector space spanned by the states in the system.

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Navier–Stokes equations

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

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Negation

In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P (¬P), which is interpreted intuitively as being true when P is false, and false when P is true.

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Nonelementary integral

In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, and logarithmic functions using field operations).

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Nonlinear optics

Nonlinear optics (NLO) is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light.

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Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation.

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Nonlinear system identification

System identification is a method of identifying or measuring the mathematical model of a system from measurements of the system inputs and outputs.

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Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.

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Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

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Pendulum (mathematics)

The mathematics of pendulums are in general quite complicated.

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Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

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Perturbation theory

Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.

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Phase portrait

A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane.

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Physicist

A physicist is a scientist who has specialized knowledge in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.

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Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Power-flow study

In power engineering, the power-flow study, or load-flow study, is a numerical analysis of the flow of electric power in an interconnected system.

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Proportionality (mathematics)

In mathematics, two variables are proportional if there is always a constant ratio between them.

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Randomness

Randomness is the lack of pattern or predictability in events.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Recurrence relation

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.

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Richards equation

The Richards equation represents the movement of water in unsaturated soils, and was formulated by Lorenzo A. Richards in 1931.

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Root-finding algorithm

In mathematics and computing, a root-finding algorithm is an algorithm for finding roots of continuous functions.

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Scale analysis (mathematics)

Scale analysis (or order-of-magnitude analysis) is a powerful tool used in the mathematical sciences for the simplification of equations with many terms.

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Science

R. P. Feynman, The Feynman Lectures on Physics, Vol.1, Chaps.1,2,&3.

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Scientist

A scientist is a person engaging in a systematic activity to acquire knowledge that describes and predicts the natural world.

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Self-balancing unicycle

A self-balancing unicycle (also electric unicycle) is a self-balancing personal transporter with a single wheel.

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Separation of variables

In mathematics, separation of variables (also known as the Fourier method) 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.

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Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

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Simple harmonic motion

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

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Sine-Gordon equation

The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function.

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Singularity (mathematics)

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

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Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave packet that maintains its shape while it propagates at a constant velocity.

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Stanislaw Ulam

Stanisław Marcin Ulam (13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics.

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Superposition principle

In physics and systems theory, 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.

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System

A system is a regularly interacting or interdependent group of items forming an integrated whole.

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System of equations

In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought.

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System of polynomial equations

A system of polynomial equations is a set of simultaneous equations f1.

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Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

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Van der Pol oscillator

In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping.

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Variable (mathematics)

In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown.

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Vector soliton

In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation.

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Vlasov equation

The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb.

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Volterra series

The Volterra series is a model for non-linear behavior similar to the Taylor series.

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References

[1] https://en.wikipedia.org/wiki/Nonlinear_system

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