an attractor

Roll a marble around inside a bowl and, whatever flick you give it, friction eventually settles it at the bottom. Set a pendulum clock swinging too softly or too hard and it relaxes into the same steady tick. There is some final behaviour the system drifts toward and then keeps doing, no matter the details of how it started, as long as it started 'nearby'. That long-run destination is an attractor.

An attractor is an invariant set A toward which all nearby trajectories converge as time goes to infinity, and which cannot be split into smaller pieces with the same property. 'Invariant' means once you are on A you stay on A; 'attracting' means there is a surrounding neighbourhood from which every trajectory approaches A. The simplest attractors are a single stable equilibrium (the marble at the bowl's bottom — the system ends motionless) and a stable limit cycle (a steady oscillation the system locks into, like the clock's regular swing). The set of all starting points that funnel into a given attractor is its basin of attraction.

Attractors are what experiments and simulations actually show you, because transient startup behaviour decays and only the attractor's behaviour persists to be observed. The dramatic discovery of chaos was that an attractor need not be a point or a loop: it can be a strange attractor, a fractal set on which the motion never repeats yet stays bounded forever. So 'the system settles down' does not always mean 'the system goes quiet' — sometimes it settles into permanent, intricate, never-repeating motion.