Closed orbit

A closed orbit returns to the same complete state and repeats its motion. For a nonconstant solution of an autonomous differential equation with unique solutions, $x(t+T)=x(t)$ for a positive period $T$ describes a periodic orbit. This usage excludes an equilibrium point, which is a stationary state rather than a moving cycle.

A harmonic oscillator has closed ellipses in its phase plane, because both position and velocity return after one oscillation. Returning to position alone is insufficient: a particle can pass the same point with a different velocity. A bound orbit also need not close; a precessing orbit can remain in a limited spatial region while changing its direction of closest approach. Specifying the complete state prevents a repeating projection from being mistaken for a repeating physical motion.