Bifurcation

A bifurcation is a sudden, qualitative change in the long-term behavior of a Dynamical System as a control parameter is varied smoothly. For most values of the parameter, nudging it a little changes the dynamics only a little. But at isolated critical values, the structure of the Phase Portrait reorganizes: a Fixed Point appears or disappears, an equilibrium loses Stability, or a steady state gives way to an oscillation. The parameter moved continuously; the behavior jumped.

The common local bifurcations

Three patterns account for most one-parameter bifurcations:

• Saddle-node — two fixed points (one stable, one unstable) collide and annihilate. Below the threshold there are two equilibria; above it, none. This is how steady states are created and destroyed.
• Pitchfork — a single stable equilibrium loses stability and gives birth to two new stable ones, symmetric about it. A ruler pressed end-on stays straight until it buckles left or right.
• Hopf — a fixed point loses stability and throws off a Limit Cycle: the system stops resting and starts oscillating. This is the birth of rhythm.

A bifurcation diagram

Plot the locations of the equilibria against the parameter r and you get a bifurcation diagram — a map of the system's repertoire. Here is the supercritical pitchfork: one branch splits cleanly into two at r=0.

Why it matters

Bifurcations are where systems change character: lasers switch on, fluids begin to convect, populations start to cycle, a heartbeat turns arrhythmic. A cascade of period-doubling bifurcations is one of the standard roads to Chaos — the Logistic Map travels exactly this route.

See also

• Logistic Map
• Stability
• Limit Cycle