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Inmathematics, specifically in the study ofdynamical systems, anorbit is a collection of points related by theevolution function of the dynamical system. It can be understood as the subset ofphase space covered by the trajectory of the dynamical system under a particular set ofinitial conditions, as the system evolves. As a 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 apartition of the phase space. Understanding the properties of orbits by usingtopological methods is one of the objectives of the modern theory of dynamical systems.

Fordiscrete-time dynamical systems, the orbits aresequences; forreal dynamical systems, the orbits arecurves; and forholomorphic dynamical systems, the orbits areRiemann surfaces.

Given a dynamical system (T,M, Φ) withT agroup,M aset and Φ the evolution function

${\displaystyle \mathrm{\Phi }:U\to M}$ where${\displaystyle U\subset T\times M}$ with${\displaystyle \mathrm{\Phi }(0,x)=x}$

we define

${\displaystyle I(x):=\{t\in T:(t,x)\in U\},}$

then the set

${\displaystyle {\gamma }_x:=\{\mathrm{\Phi }(t,x):t\in I(x)\}\subset M}$

is called theorbit throughx. An orbit which consists of a single point is calledconstant orbit. A non-constant orbit is calledclosed orperiodic if there exists a${\displaystyle t\ne 0}$ in${\displaystyle I(x)}$ such that

${\displaystyle \mathrm{\Phi }(t,x)=x}$.

Given a real dynamical system (R,M, Φ),I(x) is an open interval in thereal numbers, that is${\displaystyle I(x)=(t_x^{-},t_x^{+})}$. For anyx inM

${\displaystyle {\gamma }_x^{+}:=\{\mathrm{\Phi }(t,x):t\in (0,t_x^{+})\}}$

is calledpositive semi-orbit throughx and

${\displaystyle {\gamma }_x^{-}:=\{\mathrm{\Phi }(t,x):t\in (t_x^{-},0)\}}$

is callednegative semi-orbit throughx.

For a discrete time dynamical system with a time-invariant evolution function${\displaystyle f}$:

Theforward orbit of x is the set :

${\displaystyle {\gamma }_x^{+}\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\{f^t(x):t\ge 0\}}$

If the function is invertible, thebackward orbit of x is the set :

${\displaystyle {\gamma }_x^{-}\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\{f^t(x):t\le 0\}}$

andorbit of x is the set :

${\displaystyle {\gamma }_x\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}{\gamma }_x^{-}\cup {\gamma }_x^{+}}$

where :

• ${\displaystyle f}$ is the evolution function${\displaystyle f:X\to X}$
• set${\displaystyle X}$ is thedynamical space,
• ${\displaystyle t}$ is number of iteration, which isnatural number and${\displaystyle t\in T}$
• ${\displaystyle x}$ is initial state of system and${\displaystyle x\in X}$

For a general dynamical system, especially in homogeneous dynamics, when one has a "nice" group${\displaystyle G}$ acting on a probability space${\displaystyle X}$ in a measure-preserving way, an orbit${\displaystyle G.x\subset X}$ will be called periodic (or equivalently, closed) if the stabilizer${\displaystyle Sta{b}_G(x)}$ is a lattice inside${\displaystyle G}$.

In addition, a related term is a bounded orbit, when the set${\displaystyle G.x}$ is pre-compact inside${\displaystyle X}$.

The classification of orbits can lead to interesting questions with relations to other mathematical areas, for example the Oppenheim conjecture (proved by Margulis) and the Littlewood conjecture (partially proved by Lindenstrauss) are dealing with the question whether every bounded orbit of some natural action on the homogeneous space${\displaystyle S{L}_3(\mathbb{R})\mathrm{\setminus }S{L}_3(\mathbb{Z})}$ is indeed periodic one, this observation is due to Raghunathan and in different language due to Cassels and Swinnerton-Dyer . Such questions are intimately related to deep measure-classification theorems.

It is often the case that the evolution function can be understood to compose the elements of agroup, in which case thegroup-theoretic orbits of thegroup action are the same thing as the dynamical orbits.

• 
  Critical orbit of discrete dynamical system based oncomplex quadratic polynomial. It tends to weaklyattractingfixed point with multiplier=0.99993612384259
• 
  Critical orbit tends to weakly attracting point. One can see spiral from attracting fixed point to repelling fixed point ( z= 0) which is a place with high density of level curves.

• The orbit of anequilibrium point is a constant orbit.

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 ways. It could be anasymptotically periodic orbit if itconverges 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 bechaotic. These orbits come arbitrarily close to the initial point, but fail to ever converge to a periodic orbit. They exhibitsensitive 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 behyperbolic if nearby points approach or diverge from the orbit exponentially fast.

• Wandering set
• Phase space method
• Phase space
• Cobweb plot or Verhulst diagram
• Periodic points of complex quadratic mappings and multiplier of orbit
• Orbit portrait

• Hale, Jack K.; Koçak, Hüseyin (1991). "Periodic Orbits".Dynamics and Bifurcations. New York: Springer. pp. 365–388.ISBN 0-387-97141-6.
• Katok, Anatole; Hasselblatt, Boris (1996).Introduction to the modern theory of dynamical systems. Cambridge.ISBN 0-521-57557-5.
• Perko, Lawrence (2001)."Periodic Orbits, Limit Cycles and Separatrix Cycles".Differential Equations and Dynamical Systems (Third ed.). New York: Springer. pp. 202–211.ISBN 0-387-95116-4.

• Dynamical systems
• Group actions

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