saddle-node bifurcation
Two fixed points, one stable and one unstable, drift toward each other as you turn a parameter, meet, and then vanish together, leaving no equilibrium at all where there had been two. This mutual annihilation (or, run backward, creation out of nothing) is the saddle-node bifurcation, the most basic way for equilibria to appear or disappear.
Its normal form is dx/dt = r + x^2 (or, with the opposite sign convention, r - x^2). Look at dx/dt = r - x^2: for r > 0 there are two fixed points at x* = +sqrt(r) and x* = -sqrt(r), one stable and one unstable; at r = 0 they merge into a single half-stable point at the origin; for r < 0 there are none. The merging happens exactly when both f(x*) = 0 and f'(x*) = 0, that is, when the curve f(x) becomes tangent to the axis, which is why one eigenvalue passes through zero at the transition, the signature of this bifurcation. In two dimensions the colliding points are literally a saddle and a node, giving the bifurcation its name.
You meet the saddle-node whenever a state simply ceases to exist as a parameter is pushed too far: the onset of neural spiking, the sudden extinction of a limit cycle, and, more dramatically, tipping points and hysteresis in climate or ecological models, where crossing the fold makes a stable state disappear with no smooth way back. The name blue-sky captures the idea that a pair of fixed points can appear seemingly out of a clear blue sky.
In dx/dt = r - x^2, sweeping r downward through 0 makes the stable point at +sqrt(r) and the unstable point at -sqrt(r) rush together and annihilate; below r = 0 the state they represented no longer exists.
Two equilibria collide and vanish: a system can lose a stable state with no smooth return.
At a saddle-node the merging fixed point is not exponentially attracting; near the threshold the system creeps agonizingly slowly through the region where the points used to be, a hallmark ghost or bottleneck that a purely linear analysis, giving a zero eigenvalue, cannot capture.