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. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is, different orbits do not intersect in 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

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 ${\displaystyle t\neq 0}$ in ${\displaystyle I(x)}$ such that

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