bifurcation
/ by-fur-KAY-shun /
Turn up the heat under a pan of water slowly and for a while nothing much changes, then, at a sharp threshold, smooth conduction gives way to churning convection rolls. A control knob was turned smoothly, yet the behavior changed abruptly and qualitatively. A bifurcation is exactly this: a value of a parameter at which the very structure of a system's long-term behavior suddenly reorganizes.
Formally, consider a family of systems dx/dt = f(x, r) depending on a control parameter r. A bifurcation occurs at a parameter value r_c where the number or the stability of the fixed points (or limit cycles) changes as r passes through r_c. Since linear stability is governed by the eigenvalues of the Jacobian, bifurcations happen precisely when an eigenvalue crosses the imaginary axis, its real part passing through zero, so that the linear test goes marginal and the topology of the phase portrait is forced to change. Diagrams that plot the location and stability of these attractors against r are called bifurcation diagrams.
Bifurcations are the vocabulary of how complex behavior is born from simple equations: the basic local types, saddle-node, transcritical, pitchfork, and Hopf, are classified by their normal forms, the simplest equation exhibiting each. A cascade of one particular type, period doubling, is a standard route to chaos. The word bifurcation literally means splitting in two, which fits the pitchfork case, but the term is now used for any qualitative transition, including ones where fixed points are destroyed rather than created.
In the logistic map, as r increases the single stable state first splits into a 2-cycle near r = 3, then a 4-cycle, then 8, 16, ..., an accelerating cascade of bifurcations that piles up at the onset of chaos.
Each fork in the bifurcation diagram marks a value of r where the system's rhythm reorganizes.
A bifurcation is a change of qualitative structure, not merely a large quantitative change; a parameter can be varied enormously with no bifurcation, and a bifurcation can be triggered by an infinitesimal change right at the threshold.