a limit cycle
Some systems do not settle to a resting point and do not blow up either — they fall into a steady, repeating rhythm all on their own. A heartbeat, a firing neuron, the tick of certain electronic oscillators: each returns to the very same cycle even after you disturb it. A limit cycle is the mathematical name for such a self-chosen rhythm — an isolated closed loop in the phase plane that nearby trajectories spiral toward or away from.
The key word is isolated. A closed orbit is a trajectory that loops back on itself, giving periodic motion. In a center (like an undamped oscillator) there is a whole nested family of closed orbits, one through every nearby point — none is special. A limit cycle is different: it is a single closed orbit with no other closed orbits immediately around it, so neighbouring trajectories cannot be closed; they must spiral. If they spiral inward from both sides, the cycle is stable (an attractor); if outward, unstable; if inward on one side and outward on the other, semistable. A stable limit cycle is the hallmark of a genuinely nonlinear system — linear systems can have centers, but never an isolated, robust periodic orbit.
Limit cycles are everywhere self-sustained oscillation lives: the van der Pol oscillator, predator-prey cycles, the beating of the heart, glycolysis, and engineered clocks. They cannot occur in a linear constant-coefficient system, nor in a gradient system, nor (by Bendixson's criterion) where the flow's divergence keeps one sign. The Poincare-Bendixson theorem is the main tool for proving one exists: trap trajectories in a ring-shaped region with no equilibrium inside, and a limit cycle must live there.
In polar coordinates r' = r(1 - r^2), theta' = 1, the circle r = 1 is a stable limit cycle: inside (r < 1) r grows toward 1, outside (r > 1) it shrinks toward 1, so all nonzero trajectories spiral onto the unit circle.
Trajectories from both sides spiral onto r = 1, the isolated closed orbit — the signature of a stable limit cycle.
A center is NOT a limit cycle: a center's closed orbits come in a continuous family and are not isolated, and they are fragile (a small perturbation destroys them). A true limit cycle is isolated and robust, which is why only nonlinear systems have them.