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Concept
Orbit (dynamics)
Summary
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
where with
we define
then the set
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 in such that
Given a real dynamical system (R, M, Φ), I(x) is an open interval in the real numbers, that is . For any x in M
is called positive semi-orbit through x and
is called negative semi-orbit through x.
For discrete time dynamical system :
forward orbit of x is a set :
backward orbit of x is a set :
and orbit of x is a set :
where :
is an evolution function which is here an iterated function,
set is dynamical space,
is number of iteration, which is natural number and
is initial state of system and
Usually different notation is used :
is written as
where is in the above notation.
For a general dynamical system, especially in homogeneous dynamics, when one has a "nice" group acting on a probability space in a measure-preserving way, an orbit will be called periodic (or equivalently, closed) if the stabilizer is a lattice inside .
Official source
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