Orbit (dynamics)

In the study of dynamical systems, an orbit is a collection of points related by time evolution. The points of the orbit will be a subset of the phase or state space of the dynamical system. If the dynamical system is a map, the orbit is a sequence and if the dynamical system is a flow, the orbit is a curve. Understanding the properties of orbits is one of the objectives of the modern geometrical theory of dynamical systems.

If x is a point on the orbit, then the evolution function of the dynamical system, f ^t relates that initial point to the other points of the orbit: if y is on the orbit, then there is a value of t such that either y = f ^t(x) or x = f ^t(y). Both solutions occur when the dynamical system is reversible. For maps (or discrete-time dynamical systems) t is an integer and for flows (continuous-time dynamical systems) t is a real number. 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.

An orbit is called closed if a point of the orbit evolves to itself. This means that the orbit will repeat itself. Such orbits are also called periodic. The simplest closed orbit is a fixed point, where the orbit is a single point.

Design or Reference Orbit

The reference orbit consists of a series of straight line segments and circular arcs. It is defined under the assumption that all elements are perfectly aligned along the design orbit. The accompanying tripod of the reference orbit spans a local curvilinear right handed coordinate system, (x,y,s) The local -axis is the tangent to the reference orbit. The two other axes are perpendicular to the reference orbit and are labelled  (in the bend plane) and y (perpendicular to the bend plane).

Closed Orbit Distortion (COD)

Due to various errors like misalignment errors, field errors, fringe fields etc., the closed orbit does not coincide with the design orbit. It also changes with the momentum error. The closed orbit is described with respect to the reference orbit, using the local reference system. It is evaluated including any nonlinear effects. The resulted orbit is also called as perturbed orbit.

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