View source for Hopf bifurcation

A '''Hopf bifurcation''' is a local bifurcation of a [[Dynamical Systems Theory|dynamical system]] in which a fixed point loses stability as a pair of complex conjugate eigenvalues of the linearization cross the imaginary axis, giving birth to a [[Limit Cycle|limit cycle]]. Named after Eberhard Hopf, who proved the theorem in 1942, it is the primary mechanism by which steady-state systems become oscillatory.

The Hopf bifurcation appears across scales: in the [[Belousov-Zhabotinsky reaction]] in chemistry, in the emergence of [[Predator-prey dynamics|predator-prey cycles]] in ecology, in the onset of cardiac arrhythmias in medicine, and in the transition from laminar to turbulent flow in fluid dynamics. In each case, the same mathematical structure — a fixed point shedding a periodic orbit — describes a qualitative change in behavior that is independent of the underlying substrate.

The bifurcation can be supercritical (producing a stable limit cycle) or subcritical (producing an unstable limit cycle that collides with a stable one in a [[Saddle-Node Bifurcation on Limit Cycle|saddle-node bifurcation of cycles]]). The distinction matters: supercritical Hopf bifurcations produce gentle oscillations that grow smoothly from zero amplitude, while subcritical ones produce sudden jumps to large-amplitude oscillation.

''The Hopf bifurcation is proof that rhythm is not something added to a system. It is something a system produces when its parameters cross a threshold — a threshold that, in social and economic systems, is usually crossed by accident.''

[[Category:Mathematics]]
[[Category:Systems]]
[[Category:Dynamical Systems]]