Saddle-Node Bifurcation
Change one parameter to create or remove equilibria and inspect the different flow on each side of the critical point.
Free theory articleChange one parameter to create or remove equilibria and inspect the different flow on each side of the critical point.
Concept & interpretation
A saddle-node (or fold) bifurcation is the canonical local mechanism by which two equilibria of a one-dimensional autonomous dynamical system emerge or collide and annihilate as a control parameter crosses a critical threshold. Its normal form is dx/dt = f(x; r) = r - x2 with Jacobian derivative f'(x) = -2x.
Interactive Experiment
Trace the creation and collision of equilibria across the bifurcation point in the Saddle-Node Bifurcation interactive workspace:
- Set Parameter r = -1 and Probe x = 0. Verify Equilibrium count = 0, both equilibria read Unavailable, and Probe velocity = -1 drives flow to the Left everywhere.
- Switch Parameter r = 0 and compare Probe x = 1 versus Probe x = -1. In both probes, Probe velocity = -1 (flow is Left), showing that the single equilibrium at
x = 0attracts from the right (x > 0) and repels toward-∞on the left (x < 0), making it One semistable equilibrium. - Set Parameter r = 1 with Probe x = 0 (between the roots, flowing Right with velocity
+1) and then Probe x = 1 (sitting at the stable root with Probe velocity = 0 and Probe derivative = -2). - Set Parameter r = 4 and Probe x = 3 (outside the stable root
x = 2) to observe restoring leftward velocity4 - 32 = -5and derivative-6.