Orbit (dynamics)

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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.

[edit] Closed orbits

An orbit can fail to be closed in two interesting 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.