sink, source, and semistable points

Equilibria of a first-order autonomous equation come in three flavours, and you can tell them apart just by how the arrows on the phase line point near each one. Think of solutions as water flowing along the line: a sink is a drain everything pours into, a source is a spring everything flows out of, and a semistable point is a one-sided trap that catches flow from one direction but lets it escape on the other.

Precisely, at an equilibrium y* of dy/dt = f(y): a sink (stable equilibrium) has arrows on both sides pointing toward it — f > 0 just below and f < 0 just above — so every nearby solution converges to it. A source (unstable equilibrium) has arrows on both sides pointing away — f < 0 below and f > 0 above — so nearby solutions flee. A semistable point has both arrows pointing the same way, in on one side and out the other (for instance f > 0 on both sides): solutions approach from one direction but are repelled on the other, so it attracts some neighbours and repels others. Semistable points appear exactly when f touches zero without crossing it, like a double root, f(y) = (y − y*)^2.

These three labels are the vocabulary of long-term behaviour. Sinks are the states a system actually settles into and the ones you observe; sources are the tipping points it balances on only in theory; semistable points are delicate borderline cases that often signal an impending bifurcation — nudge a parameter and a semistable equilibrium typically splits into a sink-source pair or vanishes entirely. Reading off which is which is the payoff of the whole phase-line method.