# 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. 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.

## Definition Diagram showing the periodic orbit of a mass-spring system in simple harmonic motion. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)

Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function

$\Phi :U\to M$ where $U\subset T\times M$ with $\Phi (0,x)=x$ we define

$I(x):=\{t\in T:(t,x)\in U\},$ then the set

$\gamma _{x}:=\{\Phi (t,x):t\in I(x)\}\subset M$ 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 $t\neq 0$ in $I(x)$ such that

$\Phi (t,x)=x$ .

### Real dynamical system

Given a real dynamical system (R, M, Φ), I(x) is an open interval in the real numbers, that is $I(x)=(t_{x}^{-},t_{x}^{+})$ . For any x in M

$\gamma _{x}^{+}:=\{\Phi (t,x):t\in (0,t_{x}^{+})\}$ is called positive semi-orbit through x and

$\gamma _{x}^{-}:=\{\Phi (t,x):t\in (t_{x}^{-},0)\}$ is called negative semi-orbit through x.

### Discrete time dynamical system

For discrete time dynamical system :

forward orbit of x is a set :

$\gamma _{x}^{+}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{\Phi (t,x):t\geq 0\}$ backward orbit of x is a set :

$\gamma _{x}^{-}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{\Phi (-t,x):t\geq 0\}$ and orbit of x is a set :

$\gamma _{x}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \gamma _{x}^{-}\cup \gamma _{x}^{+}$ where :

• $\Phi$ is an evolution function $\Phi :X\to X$ which is here an iterated function,
• set $X$ is dynamical space,
• $t$ is number of iteration, which is natural number and $t\in T$ • $x$ is initial state of system and $x\in X$ Usually different notation is used :

• $\Phi (t,x)$ is written as $\Phi ^{t}(x)$ • $x_{t}=\Phi ^{t}(x)$ where $x_{0}$ is $x$ in the above notation.

### General dynamical system

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

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

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 $SL_{3}(\mathbb {R} )\backslash SL_{3}(\mathbb {Z} )$ 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.