Orbit (dynamics) 252250 160026542 2007-09-24T14:59:17Z 24.188.2.30 In [[mathematics]], in the study of [[dynamical system]]s, an '''orbit''' is a collection of points related by the [[evolution function]] of the dynamical system. The orbit is a subset of the [[Phase space (dynamical system)|phase space]] and the set of all orbits is a [[partition (set theory)|partition]] of the phase space, that is different orbits do not intersect in the phase space. Understanding the properties of orbits by using topological method is one of the objectives of the modern theory of dynamical systems. For [[discrete-time dynamical system]]s the orbits are [[sequence]]s, for [[real dynamical system]]s the orbits are [[curve]]s and for [[holomorphic dynamical system]]s the orbits are [[Riemann surface]]s. == Definition == Given a dynamical system (''T'', ''M'', Φ) with ''T'' a group, M a set and Φ the evolution function :<math>\Phi: T \times M \to M</math> we define :<math>I(x):=\{t \in T : (t,x) \in T \times M \},</math> then the set :<math>\gamma_x:=\{\Phi(t,x) : t \in I(x)\}</math> is called '''orbit''' through ''x''. An orbit which consists of a single point is called '''constant orbit'''. A non-constant orbit is called '''closed''' or '''periodic''' if there exists a ''t'' in ''T'' so that :<math>\Phi(t, x) = x</math> for every point ''x'' on the orbit. === [[Real dynamical system]] === Given a real dynamical system (''R'', ''M'', Φ), ''I''(''x'') is an open interval in the [[real number]]s, that is <math>I(x) = ]t_x^- , t_x^+[</math>. For any ''x'' in ''M'' :<math>\gamma_{x}^{+} := \{\Phi(t,x) : t \in ]0,t_x^+[\}</math> is called '''positive semi-orbit''' through ''x'' and :<math>\gamma_{x}^{-} := \{\Phi(t,x) : t \in ]t_x^-,0[\}</math> is called '''negative semi-orbit''' through ''x''. === Notes === It is often the case that the evolution function can be understood to compose the elements of a [[group (mathematics)|group]], in which case the [[orbit (group theory)|group-theoretic orbits]] of the [[group action]] are the same thing as the dynamical orbits. === Examples === * The orbit of a [[equilibrium point]] is a constant orbit == Stability of orbits == A basic classification of orbits is * constant orbits or fixed points * periodic orbits * non-constant and non-periodic orbits An orbit can fail to be closed in two interesting ways. It could be an '''asymptotically periodic''' orbit if it [[limit (mathematics)|converges]] to a periodic orbit. Such orbits are not closed because they never truly repeat, but they become arbitrarily close to a repeating orbit. An orbit can also be [[chaos theory|chaotic]]. These orbits come arbitrarily close to the initial point, but fail to ever converge to a periodic orbit. They exhibit [[sensitive dependence on initial conditions]], meaning that small differences in the initial value will cause large differences in future points of the orbit. There are other properties of orbits that allow for different classifications. An orbit can be [[hyperbolic (dynamical systems)|hyperbolic]] if nearby points approach or diverge from the orbit exponentially fast. ==See also== * [[Wandering set]] * [[Phase space method]] ==References== * {{cite book | author=Anatole Katok and Boris Hasselblatt | title= Introduction to the modern theory of dynamical systems | publisher= Cambridge | year= 1996 | id=ISBN 0-521-57557-5}} [[Category:Dynamical systems]] [[Category:Group actions]] [[it:Orbita (matematica)]]