This is a list of

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

and

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

topics, by Wikipedia page. See also

list of partial differential equation topics A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby unio ...

, list of equations.

Dynamical systems, in general

Deterministic system (mathematics) In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given st ...

Linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstracti ...

Partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

Dynamical systems and chaos theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called '' ...

Chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have ...

** Chaos argument ** Butterfly effect ** 0-1 test for chaos *

Bifurcation diagram In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the sy ...

* Feigenbaum constant *

Sharkovskii's theorem In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the ...

Attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...

Strange nonchaotic attractor In mathematics, a strange nonchaotic attractor (SNA) is a form of attractor which, while converging to a limit, is strange, because it is not piecewise differentiable, and also non-chaotic, in that its Lyapunov exponents are non-positive. SNAs we ...

Stability theory In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial diffe ...

Mechanical equilibrium In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is ze ...

** Astable ** Monostable **

Bistability In a dynamical system, bistability means the system has two stable equilibrium states. Something that is bistable can be resting in either of two states. An example of a mechanical device which is bistable is a light switch. The switch leve ...

Metastability In chemistry and physics, metastability denotes an intermediate energetic state within a dynamical system other than the system's state of least energy. A ball resting in a hollow on a slope is a simple example of metastability. If the ball i ...

Feedback Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...

Negative feedback Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by othe ...

Positive feedback Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop which exacerbates the effects of a small disturbance. That is, the effects of a perturbation on a system include an increase in th ...

Homeostasis In biology, homeostasis (British English, British also homoeostasis) Help:IPA/English, (/hɒmɪə(ʊ)ˈsteɪsɪs/) is the state of steady internal, physics, physical, and chemistry, chemical conditions maintained by organism, living systems. Thi ...

Damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...

Dissipative system A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dis ...

Spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...

Turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...

Perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...

Control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...

** Non-linear control **

Adaptive control Adaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumpt ...

** Hierarchical control **

Intelligent control Intelligent control is a class of control techniques that use various artificial intelligence computing approaches like neural networks, Bayesian probability, fuzzy logic, machine learning, reinforcement learning, evolutionary computation and genet ...

** Optimal control **

Dynamic programming Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. I ...

Robust control In control theory, robust control is an approach to controller design that explicitly deals with uncertainty. Robust control methods are designed to function properly provided that uncertain parameters or disturbances are found within some (typicall ...

** Stochastic control *

System dynamics System dynamics (SD) is an approach to understanding the nonlinear behaviour of complex systems over time using stocks, flows, internal feedback loops, table functions and time delays. Overview System dynamics is a methodology and mathematic ...

system analysis System analysis in the field of electrical engineering characterizes electrical systems and their properties. System analysis can be used to represent almost anything from population growth to audio speakers; electrical engineers often use it b ...

* Takens' theorem * Exponential dichotomy * Liénard's theorem *

Krylov–Bogolyubov theorem In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of ...

* Krylov-Bogoliubov averaging method

Abstract dynamical systems

Measure-preserving dynamical system In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ca ...

Ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

Mixing (mathematics) In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''. The concept appear ...

Almost periodic function In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Hara ...

Symbolic dynamics In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics ( ...

Time scale calculus In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying ...

Arithmetic dynamics Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is ...

Sequential dynamical system Sequential dynamical systems (SDSs) are a class of graph dynamical systems. They are discrete dynamical systems which generalize many aspects of for example classical cellular automata, and they provide a framework for studying asynchronous proce ...

Graph dynamical system In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate their structural propertie ...

* Topological dynamical system

Dynamical systems, examples

* List of chaotic maps *

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

Lorenz attractor The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the ...

* Lorenz-96 *

Iterated function system In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981. IFS fractals ...

Tetration In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as rep ...

Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total ...

Horseshoe map In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior ...

Hénon map The Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (''xn'', ''yn'' ...

Arnold's cat map In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name. Thinking of the torus \mathbb^2 as the quotient space ...

Population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...

Complex dynamics Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General Montel's theorem Po ...

Fatou set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values w ...

Julia set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values w ...

Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This ...

Difference equations

Recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramete ...

Matrix difference equation A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the eq ...

Rational difference equation A rational difference equation is a nonlinear difference equation of the form : x_ = \frac~, where the initial conditions x_, x_,\dots, x_ are such that the denominator never vanishes for any . First-order rational difference equation A first-orde ...

Ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
s: general

Examples of differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

Autonomous system (mathematics) In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invari ...

Picard–Lindelöf theorem In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the C ...

Peano existence theorem In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees t ...

* Carathéodory existence theorem *

Numerical ordinary differential equations Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also ...

* Bendixson–Dulac theorem *

Gradient conjecture In mathematics, the gradient conjecture, due to René Thom (1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka (University of Savoie, France), Tadeusz Mostowski ( Warsaw University, Poland) and Adam Parusiński ( University ...

Recurrence plot In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits rou ...

Limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinit ...

Initial value problem In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or o ...

* Clairaut's equation *

Singular solution A singular solution ''ys''(''x'') of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the so ...

Poincaré–Bendixson theorem In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. Theorem Given a differentiable real dynamical system defined on an o ...

Riccati equation In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form : y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2(x ...

s *

Functional differential equation A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values. Functiona ...

Linear differential equation 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 :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ ...
s

Exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...

Malthusian catastrophe Malthusianism is the idea that population growth is potentially exponential while the growth of the food supply or other resources is linear, which eventually reduces living standards to the point of triggering a population die off. This event, c ...

* Exponential response formula *

Simple harmonic motion In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...

** Phasor (physics) **

RLC circuit An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components ...

Resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...

*** Impedance *** Reactance ***

Musical tuning In music, there are two common meanings for tuning: * Tuning practice, the act of tuning an instrument or voice. * Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases. Tuning practice Tuni ...

***

Orbital resonance In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relations ...

***

Tidal resonance In oceanography, a tidal resonance occurs when the tide excites one of the resonant modes of the ocean. The effect is most striking when a continental shelf is about a quarter wavelength wide. Then an incident tidal wave can be reinforce ...

* Oscillator **

Harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...

Electronic oscillator An electronic oscillator is an electronic circuit that produces a periodic, oscillating electronic signal, often a sine wave or a square wave or a triangle wave. Oscillators convert direct current (DC) from a power supply to an alternating cur ...

Floquet theory Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form :\dot = A(t) x, with \displaystyle A(t) a piecewise continuous periodic functio ...

Fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...

Oscillation Oscillation is the repetitive or Periodic function, periodic variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. Familiar examples o ...

(Vibration) *

Fundamental matrix (linear differential equation) In mathematics, a fundamental matrix of a system of ''n'' homogeneous linear ordinary differential equations \dot(t) = A(t) \mathbf(t) is a matrix-valued function \Psi(t) whose columns are linearly independent solutions of the system. Then ev ...

Laplace transform applied to differential equations In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial condit ...

Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form: for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...

Wronskian In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian o ...

* Loewy decomposition

Mechanics

Pendulum A pendulum is a weight suspended from a wikt:pivot, pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, Mechanical equilibrium, equilibrium position, it is subject to a restoring force due to gravity that ...

Inverted pendulum An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. It can be suspended stably in this inverted position by using a control system to monitor the a ...

Double pendulum In physics and mathematics, in the area of dynamical systems, a double pendulum also known as a chaos pendulum is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a ...

Foucault pendulum The Foucault pendulum or Foucault's pendulum is a simple device named after French physicist Léon Foucault, conceived as an experiment to demonstrate the Earth's rotation. A long and heavy pendulum suspended from the high roof above a circular a ...

** Spherical pendulum * Kinematics *

Equation of motion 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.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...

Dynamics (mechanics) Dynamics is the branch of classical mechanics that is concerned with the study of forces and their effects on motion. Isaac Newton was the first to formulate the fundamental physical laws that govern dynamics in classical non-relativistic phys ...

Classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...

* Isolated physical system **

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...

Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momen ...

Routhian mechanics alt= In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the ...

** Hamilton-Jacobi theory **

Appell's equation of motion In classical mechanics, Appell's equation of motion (aka the Gibbs–Appell equation of motion) is an alternative general formulation of classical mechanics described by Josiah Willard Gibbs in 1879 and Paul Émile Appell in 1900. Statement ...

** Udwadia–Kalaba equation **

Celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

Orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...

Lagrange point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of t ...

** Kolmogorov-Arnold-Moser theorem **

N-body problem In physics, the -body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some histo ...

many-body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...

Ballistics Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing ...

Functions defined via an ODE

Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent soluti ...

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...

* Hypergeometric function

Rotating systems

Angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

Angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...

Angular acceleration In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceler ...

Angular displacement Angular displacement of a body is the angle (in radians, degrees or revolutions) through which a point revolves around a centre or a specified axis in a specified sense. When a body rotates about its axis, the motion cannot simply be analyzed ...

Rotational invariance In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. Mathematics Functions For example, the function :f(x,y) = ...

Rotational inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...

Torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of t ...

Rotational energy Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the o ...

Centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...

Centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel ...

Centrifugal governor A centrifugal governor is a specific type of governor with a feedback system that controls the speed of an engine by regulating the flow of fuel or working fluid, so as to maintain a near-constant speed. It uses the principle of proportional c ...

Coriolis force In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...

* Axis of rotation *

Flywheel A flywheel is a mechanical device which uses the conservation of angular momentum to store rotational energy; a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. In particular, assu ...

Flywheel energy storage Flywheel energy storage (FES) works by accelerating a rotor (flywheel) to a very high speed and maintaining the energy in the system as rotational energy. When energy is extracted from the system, the flywheel's rotational speed is reduced as a c ...

Momentum wheel A reaction wheel (RW) is used primarily by spacecraft for three-axis attitude control, and does not require rockets or external applicators of torque. They provide a high pointing accuracy, and are particularly useful when the spacecraft must be ...

Spinning top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will usually wobble for a f ...

* Gyroscope *

Gyrocompass A gyrocompass is a type of non-magnetic compass which is based on a fast-spinning disc and the rotation of the Earth (or another planetary body if used elsewhere in the universe) to find geographical direction automatically. The use of a gyroc ...

Precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...

Nutation Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference fra ...

Swarms

* Particle swarm optimization * Self-propelled particles *

Swarm intelligence Swarm intelligence (SI) is the collective behavior of decentralized, self-organized systems, natural or artificial. The concept is employed in work on artificial intelligence. The expression was introduced by Gerardo Beni and Jing Wang in 1989, ...

Stochastic dynamic equations

Random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb ...

* Autoregressive process *

Unit root In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if ...

* Moving average process *

Autoregressive–moving-average model In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second ...

Autoregressive integrated moving average In statistics and econometrics, and in particular in time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model. Both of these models are fitted to time ...

Vector autoregressive model Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single-variable (univariate) autoregres ...

Stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock ...

Stochastic partial differential equation Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They hav ...

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