a repeller
Balance a pencil perfectly on its point and, in principle, it could stand there forever — but the tiniest nudge sends it toppling away, and it never comes back to balance on its own. A repeller is the mathematical version of that knife-edge state: a set the system can sit on exactly, but flees from the moment it is perturbed. It is an attractor with time running backward.
Formally, a repeller is an invariant set such that every nearby trajectory moves away from it as time increases — equivalently, it is an attractor for the reversed flow phi_(-t). The plainest example is an unstable equilibrium, like the upside-down balance point of a pendulum or the source of a planar system from which all nearby trajectories spiral or shoot outward. On the line, a fixed point x* of x' = f(x) is a repeller when f'(x*) > 0, so that the rate of departure grows with displacement and small deviations are amplified rather than damped.
Repellers matter precisely because you almost never see them in a real experiment: any noise, any imprecision, knocks the system off and it leaves, so repellers are invisible in the long run yet they organize the flow. They mark the watersheds between basins of attraction — the boundary you must straddle exactly to avoid sliding into one attractor or the other — and reversing time turns the study of repellers into the study of attractors, so the two ideas are mirror images of one machine.
For x' = x - x^3, the fixed point x = 0 has f'(0) = 1 > 0, so it is a repeller: start at 0.01 and the trajectory grows toward x = 1, start at -0.01 and it falls toward x = -1. The point x = 0 is the divide between the two basins, and only an exact start of 0 stays there.
An unstable fixed point that any small displacement flees — visible only as the boundary separating where trajectories end up.
Do not confuse a repeller with a saddle: a saddle repels along some directions but attracts along others, so it is neither a pure repeller nor an attractor. A repeller pushes away in every nearby direction.