Bifurcations

Level SpecializationDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Qualitative changes in behaviour as a parameter crosses a critical value: an equilibrium loses stability, a cycle is born, periods double on the road to chaos. Tipping points in climate and ecology are bifurcations.

Formulas

$x_{k+1} = r\,x_k(1 - x_k), \qquad r_1 = 3,\ r_2 = 1 + \sqrt6,\ \dots,\ r_\infty \approx 3.5699$
logistic map: period doubling cascade

Where it shows up in computing

Population and epidemic models★★★★★frequentPhysics and simulation
Discrete population models show period doubling and chaos as the growth rate increases (May, 1976).
Control theory★★★★★advancedRobotics and control
Hopf bifurcations explain how increasing a gain turns a stable loop into an oscillating one.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚙ Robotics and control

Control theory★★★★★
Control theory→Trajectory optimization and MPC★★★★★

⚛ Physics and simulation

Population and epidemic models★★★★★

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.