Saddle-Node Bifurcation
Change one parameter to create or remove equilibria and inspect the different flow on each side of the critical point.
By Riano · Published · Updated · Report a correction
Saddle-Node Bifurcation
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.
Worked Example: Equilibrium Pair and Central Flow at r = 1
Analyze the equilibria, linear stability, and local phase-line velocity for dx/dt = r - x2 at r = 1 and probe x = 0:
- Equilibria: Solve
1 - x2 = 0to obtain two roots:x- = -1(lower equilibrium) andx+ = +1(upper equilibrium). - Linearized stability: Evaluate
f'(x) = -2xat each root. Atx- = -1,f'(-1) = +2 > 0(unstable repeller); atx+ = +1,f'(+1) = -2 < 0(locally asymptotically stable attractor). - Probe at
x = 0: The velocity isdx/dt = 1 - 02 = 1 > 0(direction Right toward the stable root at+1), while the local slope of the vector field isf'(0) = 0.
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.
Limitations and Assumptions
At the bifurcation value r = 0, the linearization f'(0) = 0 has a zero eigenvalue and is non-hyperbolic; linear stability analysis alone cannot determine stability, and the quadratic term -x2 governs the one-sided semistable flow. Furthermore, the probe velocity dx/dt = r - x2 describes the instantaneous local phase line rather than a numerically integrated global trajectory—solutions starting left of the unstable root -√r (or anywhere when r < 0) blow up to -∞ in finite time.
References
The normal form dx/dt = r - x2 and the phase-line geometry of saddle-node bifurcations are presented in the following dynamical systems lecture notes:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Saddle-Node Bifurcation module.
Try Saddle-Node Bifurcation on Riano →