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 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 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.
Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function
\Phi:U\toM
U\subsetT x M
\Phi(0,x)=x
we define
I(x):=\{t\inT:(t,x)\inU\},
then the set
\gammax:=\{\Phi(t,x):t\inI(x)\}\subsetM
is called the 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 ≠ 0
I(x)
\Phi(t,x)=x
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 |
| + | |
| \gamma | |
| x |
:=\{\Phi(t,x):t\in
| +)\} | |
| (0,t | |
| x |
| - | |
| \gamma | |
| x |
:=\{\Phi(t,x):t\in
| -,0)\} | |
| (t | |
| x |
For a discrete time dynamical system with a time-invariant evolution function
f
The forward orbit of x is the set :
| + | |
| \gamma | |
| x |
\overset{\underset{def
If the function is invertible, thebackward orbit of x is the set :
| - | |
| \gamma | |
| x |
\overset{\underset{def
and orbit of x is the set :
\gammax \overset{\underset{def
where :
f
f:X\toX
X
t
t\inT
x
x\inX
For a general dynamical system, especially in homogeneous dynamics, when one has a "nice" group
G
X
G.x\subsetX
StabG(x)
G
In addition, a related term is a bounded orbit, when the set
G.x
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
SL3(R)\backslashSL3(Z)
It is often the case that the evolution function can be understood to compose the elements of a group, in which case the group-theoretic orbits of the group action are the same thing as the dynamical orbits.
A basic classification of orbits is
An orbit can fail to be closed in two ways. It could be an asymptotically periodic orbit if it 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 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 if nearby points approach or diverge from the orbit exponentially fast.
. Jack K. Hale . Hüseyin . Koçak . Dynamics and Bifurcations . New York . Springer . 1991 . 0-387-97141-6 . Periodic Orbits . 365–388 .