About: Orbit (dynamics)

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In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems.

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rdfs:label	Orbita (matematica) (it) 軌道 (力学系) (ja) Orbit (dynamics) (en)
rdfs:comment	力学系における軌道（きどう）とは、ある初期条件を通り、系の時間発展のルールに従って定まる状態の集合である。幾何学的には軌道は、離散力学系では相空間上の点列、連続力学系では相空間上の曲線となる。より一般的には、群作用から定まる軌道と同義である。軌道の性質を調べることが、力学系という分野の主な関心の一つである。 (ja) In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems. (en) In matematica, in particolare in geometria differenziale, un'orbita di un sistema dinamico è una traiettoria percorsa dal sistema nello spazio delle fasi, ovvero una funzione che soddisfa l'equazione che definisce il sistema dinamico stesso. Se il sistema dinamico è continuo, cioè è determinato da un'equazione differenziale ordinaria autonoma: (it)
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dcterms:subject	Dynamical systems Group actions (mathematics)
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Wikipage revision ID	1038047741 (xsd:integer)
Link from a Wikipage to another Wikipage	Holomorphic function Riemann surface Curve Dynamical system Initial condition Mathematics Equilibrium point Limit (mathematics) Complex quadratic polynomial Topological dynamics Wandering set Fixed point (mathematics) Group (mathematics) Group action (mathematics) Attractor Iterated function Hyperbolic equilibrium point Dynamical systems Chaos theory Cobweb plot Group actions (mathematics) Natural number Orbit (group theory) Real number Sequence Set (mathematics) Orbit portrait Periodic points of complex quadratic mappings Phase space method Phase space (dynamical system) Discrete-time dynamical system Partition (set theory) Real dynamical system Sensitive dependence on initial conditions Evolution function
Link from a Wikipage to an external page	https://books.google.com/books%3Fid=VFnSBwAAQBAJ&pg=PA202
sameAs	Orbit (dynamics) Orbit (dynamics) Orbit (dynamics) Orbit (dynamics) Orbit (dynamics) Orbit (dynamics)
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has abstract	In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems. For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces. (en) 力学系における軌道（きどう）とは、ある初期条件を通り、系の時間発展のルールに従って定まる状態の集合である。幾何学的には軌道は、離散力学系では相空間上の点列、連続力学系では相空間上の曲線となる。より一般的には、群作用から定まる軌道と同義である。軌道の性質を調べることが、力学系という分野の主な関心の一つである。 (ja) In matematica, in particolare in geometria differenziale, un'orbita di un sistema dinamico è una traiettoria percorsa dal sistema nello spazio delle fasi, ovvero una funzione che soddisfa l'equazione che definisce il sistema dinamico stesso. Se il sistema dinamico è continuo, cioè è determinato da un'equazione differenziale ordinaria autonoma: con un campo vettoriale differenziabile definito nello spazio delle fasi , un'orbita è una soluzione dell'equazione. Dal momento che il flusso del sistema nel punto è la soluzione quando è preso come il punto di inizio dell'evoluzione del sistema, ovvero , si ha che l'orbita passante per è talvolta scritta come l'insieme: (it)
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is rdfs:seeAlso of	Complex polygon
is Link from a Wikipage to another Wikipage of	Proofs of Fermat's little theorem Schröder's equation Bertrand's theorem Horseshoe map Dynamical system Index of physics articles (O) Integral curve Invariant subspace problem Mandelbrot set Orbit Limit cycle Limit set Pugh's closing lemma

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