Orbit (dynamics)
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.
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 we define then the set is called the orbit through x.
A non-constant orbit is called closed or periodic if there exists a
For a discrete time dynamical system with a time-invariant evolution function
: The forward orbit of x is the set : If the function is invertible, the backward orbit of x is the set : and orbit of x is the set : where : For a general dynamical system, especially in homogeneous dynamics, when one has a "nice" group
will be called periodic (or equivalently, closed) if the stabilizer
In addition, a related term is a bounded orbit, when the set
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
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 a group, in which case the group-theoretic orbits of the group action are the same thing as the dynamical orbits.
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.
