Hopf bifurcation

Some systems do not slide from one steady state to another when pushed too far; instead the steady state itself starts to ring, and a small oscillation blooms out of stillness. A tap dripping faster and faster suddenly begins to oscillate; a laser abruptly starts to pulse; a chemical mixture that sat quiet begins to cycle in color. The birth of that oscillation from a fixed point is a Hopf bifurcation.

A Hopf bifurcation occurs when a fixed point loses stability not because a single real eigenvalue crosses zero, but because a complex-conjugate pair of eigenvalues of the Jacobian crosses the imaginary axis, their real part passing through zero while their imaginary part stays nonzero. Because the imaginary part sets a rotation rate, the perturbation spirals, and at the transition a small-amplitude limit cycle is born with frequency approximately equal to that imaginary part. In the supercritical case the emerging limit cycle is stable and grows continuously, with amplitude scaling like sqrt(r - r_c); in the subcritical case an unstable cycle exists before the transition and the system jumps abruptly to a distant attractor, with hysteresis.

This is the standard mechanism by which self-sustained oscillations switch on, and it is one of the few bifurcations specific to two or more dimensions (a one-dimensional flow cannot oscillate). You meet it in the onset of the van der Pol and Belousov-Zhabotinsky oscillations, in fluid instabilities, in predator-prey cycles, and in the early stages of many routes to chaos. A crucial honesty: the supercritical case is benign and reversible, but the subcritical case is dangerous, since the amplitude jumps discontinuously and does not return when the parameter is reversed, so telling the two apart requires the nonlinear terms, not just the eigenvalue crossing.