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Dynamical system (definition)

Index Dynamical system (definition)

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. [1]

58 relations: Alexandroff extension, Ambient space, Attractor, Banach space, Cellular automaton, Classical mechanics, Compact space, Continuous function, Differentiable function, Differential equation, Discrete time and continuous time, Dissipative system, Dynamical system, Empty set, Ergodic theory, Flow (mathematics), Force, Formal system, Function (mathematics), Functional (mathematics), Graph (discrete mathematics), Graph of a function, Hamiltonian system, Hausdorff space, Image (mathematics), Initial value problem, Integer, Integer lattice, Interval (mathematics), Iterated function, Krylov–Bogolyubov theorem, Lattice (group), Lebesgue measure, Limit set, Liouville's theorem (Hamiltonian), Locally compact space, Manifold, Mathematics, Measurable function, Measure space, Measure-preserving dynamical system, Monoid, Orbit (dynamics), Ordinary differential equation, Phase space, Projection (set theory), Real number, Set (mathematics), Sigma-algebra, Simply connected space, ..., Space, Symplectic manifold, Time, Topological space, Trajectory, Tuple, Vector field, Velocity. Expand index (8 more) »

Alexandroff extension

In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.

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Ambient space

An ambient space or ambient configuration space is the space surrounding an object.

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Attractor

In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system.

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Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Cellular automaton

A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model studied in computer science, mathematics, physics, complexity science, theoretical biology and microstructure modeling.

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Classical mechanics

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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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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Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

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Differential equation

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

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Discrete time and continuous time

In mathematics and in particular mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time.

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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.

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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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Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

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Ergodic theory

Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies dynamical systems with an invariant measure and related problems.

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Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid.

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Force

In physics, a force is any interaction that, when unopposed, will change the motion of an object.

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Formal system

A formal system is the name of a logic system usually defined in the mathematical way.

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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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Functional (mathematics)

In mathematics, the term functional (as a noun) has at least two meanings.

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Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

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Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

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Hamiltonian system

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

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Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

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Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

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Initial value problem

In mathematics, the field of differential equations, an initial value problem (also called the Cauchy problem by some authors) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution.

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Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Integer lattice

In mathematics, the n-dimensional integer lattice (or cubic lattice), denoted Zn, is the lattice in the Euclidean space Rn whose lattice points are ''n''-tuples of integers.

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Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

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Iterated function

In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times.

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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.

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Lattice (group)

In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.

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Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

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Limit set

In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time.

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Liouville's theorem (Hamiltonian)

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.

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Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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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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Measurable function

In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open.

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Measure space

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes.

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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.

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Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

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Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system.

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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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Phase space

In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.

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Projection (set theory)

In set theory, a projection is one of two closely related types of functions or operations, namely.

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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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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Sigma-algebra

In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections.

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Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

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Space

Space is the boundless three-dimensional extent in which objects and events have relative position and direction.

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Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.

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Time

Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Trajectory

A trajectory or flight path is the path that a massive object in motion follows through space as a function of time.

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Tuple

In mathematics, a tuple is a finite ordered list (sequence) of elements.

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Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

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Velocity

The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time.

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Continuous-time dynamical system, Continuously differentiable real dynamical system, Differentiable dynamical system, Differentiable dynamics, Differentiable real dynamical system, Discrete dynamical system, Discrete-time dynamical system, Evolution function, Evolution parameter, Global dynamical system, Ph-invariant, Real dynamical system, Real global dynamical system, Φ-invariant.

References

[1] https://en.wikipedia.org/wiki/Dynamical_system_(definition)

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