a closed orbit

A closed orbit is a trajectory that loops back exactly onto itself — the state travels around and returns to precisely where it began, then repeats forever. It is the phase-plane picture of perfectly periodic motion: a heartbeat, a planet's orbit, an undamped pendulum swinging back and forth identically for all time. Where most trajectories drift somewhere and stop, a closed orbit is a journey with no destination, only an endless lap.

Concretely, a closed orbit is a non-equilibrium trajectory that is a closed loop in the phase plane, corresponding to a periodic solution: there is a period P such that (x(t + P), y(t + P)) = (x(t), y(t)) for all t. The state circulates around the loop, completing one full cycle every P units of time. Closed orbits come in two flavours that are easy to confuse but profoundly different. Around a center, they come in nested families — a continuum of loops, one through every nearby point, each with its own amplitude (this is what a conserved quantity produces). A limit cycle, by contrast, is an ISOLATED closed loop that neighbouring trajectories spiral toward or away from; that is a nonlinear phenomenon and belongs to the nonlinear theory, not here.

Closed orbits are how a phase portrait encodes sustained oscillation. The everyday rule for finding the nested kind is to look for a conserved quantity: if some function E(x, y) stays constant along every trajectory, then each trajectory is trapped on a level curve E = constant, and wherever those level curves close up into loops you have closed orbits encircling a center. A frictionless or otherwise loss-free system is the classic source — energy is conserved, its level sets are closed, and the motion repeats.