Orbit (dynamics)

In the study of dynamical systems, an orbit is the sequence generated by iterating a map.

An orbit is called closed if this sequence is finite. In simple terms, this means that the orbit will repeat itself. Such an orbit may be periodic, meaning that the entire sequence repeats. Othewise it is eventually periodic, meaning that the sequence will start in a non-repeating orbit but will enter a repeating orbit after some finite number of iterations. The simplest closed orbit is a fixed point, where the orbit is a single point.

If the map is on a metric space (where the concept of distance exists,) an orbit is asymptotically periodic if the orbit converges to a periodic orbit. Such orbits are not closed because they never truly repeat, but they become arbitrarily close to a repeating orbit.

The most interesting orbits are those that are chaotic. These orbits are not closed or asymptotically periodic. They also demonstrate sensitive dependence on initial conditions, meaning that small differences in the initial value will cause large differences in the subsequent orbits.