Bifurcations and the loss of stability

Fix a nonlinear system's equations and vary one parameter in them — the strength of the drive, the temperature difference across a fluid layer, the load on a column. For almost every value of that parameter, changing it slightly does nothing interesting: the fixed points slide a little and keep whatever stability they had. A bifurcation is one of the isolated values where that fails, and the qualitative structure of the solutions changes.

Three of them cover most of what a first course meets. In a saddle-node bifurcation a stable fixed point and an unstable one approach each other, merge and disappear, which is how a system can run out of a steady state entirely and be forced somewhere else. In a pitchfork, a symmetric state stays put but loses its stability to two new states related by the system's own symmetry — a compressed column is straight until the critical load and bows left or right above it, and which side it picks is decided by whatever imperfection is present, not by the equations. In a Hopf bifurcation the departing stability goes to an oscillation instead: the fixed point is still there, but the system now circles it on a small closed loop whose amplitude grows from zero as the parameter increases. Heart rhythms, chemical oscillators and flutter in an aircraft wing all start this way.

The distinction that matters in practice is supercritical versus subcritical. In the supercritical case the new state grows continuously out of the old one, so crossing the threshold is undramatic and reversible — back the parameter off and the system returns the way it came. In the subcritical case the branch that takes over is far away, so the system jumps, and on the way back it stays on the new branch past the point where it left the old one: hysteresis, and a threshold that depends on which direction you approached it from. Reading which case you are in off the diagram is what makes the difference between a design that degrades gracefully and one that fails all at once.