an invariant set
Imagine a fenced pasture set down inside the flowing river. If a leaf that starts inside the fence can never drift out — and, going backward, every leaf inside the fence must have come from inside it too — then the fence encloses a region the flow can shuffle around internally but never leak across. Such a region is an invariant set: a collection of states that the dynamics keeps to itself for all time.
Formally, a set S in phase space is invariant for the flow phi_t if phi_t(S) is contained in S for every time t — starting anywhere in S, you stay in S forever. (When this holds for both forward and backward time the set is called fully or strictly invariant.) The simplest invariant sets are single equilibrium points, which the flow never moves, and whole orbits, which the flow merely slides a state along. But invariant sets can be larger and more interesting: a closed loop the system circulates around forever (a limit cycle), an entire attractor, or a region of the plane that solutions enter and then never escape.
Invariant sets are the skeleton of the long-term picture. Because the flow can never carry a trajectory out of one, they partition phase space into self-contained 'worlds', and the whole strategy of qualitative analysis is to locate them — fixed points, periodic orbits, attractors — and understand how the flow behaves on and between them. The Poincare-Bendixson theorem, for instance, is fundamentally a statement about what invariant sets can exist in the plane.
For x' = x - x^3 on the line, each of the three equilibria x = -1, 0, 1 is an invariant set, and so is every open interval between adjacent equilibria, such as (0, 1): a point starting at 0.3 moves toward 1 but can never cross either endpoint, because the endpoints are themselves fixed and trajectories cannot collide.
Equilibria fence the line into invariant intervals — the flow rearranges points inside each but never carries one across a fixed point.
Invariance is about what the flow does, not about standing still: an invariant set can be full of vigorous motion (a whole orbit, a limit cycle). The defining property is only that nothing ever enters or leaves, not that nothing moves.