the basin of attraction
A stable equilibrium pulls nearby states toward itself — but how near is 'nearby'? The basin of attraction answers that. It is the full set of starting points from which the system eventually ends up at a given attractor. Picture rainfall on a landscape: every drop that lands within one valley drains to that valley's lowest pond. The whole catchment area feeding one pond is its basin of attraction; the ridges between catchments are the boundaries.
Precisely, for an attractor (a stable equilibrium, or a stable closed orbit), its basin is the collection of all initial states (x0, y0) whose trajectory converges to that attractor as time goes to infinity. If a system has several attractors, the phase plane is partitioned into their basins, and the boundaries between basins are special separating curves — often the stable manifold of a saddle point acts as exactly such a watershed, routing states on one side to one attractor and states on the other to a different one. A single basin can be a tidy disk, or a strangely shaped, even fractal, region.
Basins matter because stability is local but consequences are global: an equilibrium being stable only promises that states close enough are captured, and the basin tells you HOW close is close enough — the safety margin. In engineering this is the difference between a perturbation the system recovers from and one that tips it into a different, perhaps catastrophic, attractor. Knowing an equilibrium is stable is not enough; you must know the size and shape of its basin to know whether a real disturbance stays caught.
In a system with two stable equilibria separated by a saddle, the saddle's stable manifold splits the plane in two: every start on the left drains to the left equilibrium, every start on the right drains to the right one. Those two half-regions are the two basins.
A saddle's stable manifold is the watershed dividing one basin of attraction from the next.
Stability is local; a basin is global. An equilibrium can be perfectly stable yet have a tiny basin, so a modest disturbance still escapes it and lands in a different attractor. Calling something stable says nothing about how big a kick it can survive — that is precisely what the basin measures.