Normal form
The simplest saddle-node model has one control parameter.
dx/dt=r+x²
Saddle-Node Bifurcation
From smooth parameter change to sudden state loss
A saddle-node bifurcation occurs when a stable and unstable equilibrium collide and disappear.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: a continuously changing parameter can abruptly eliminate the state a system has been following.
One Concept · One Example
Saddle-Node Bifurcation
Presented by Charlotte Moser · Coming soon
What to notice
- For r<0, two equilibria exist.
- At r=0, the branches meet.
- For r>0, no equilibrium remains.
The mathematical structure
Equilibria
Real branches exist only before the threshold.
x*=±√(−r), r≤0
Approach a tipping threshold
Move r through zero and connect the local phase line to the full bifurcation diagram.
At a saddle-node bifurcation, a stable state can vanish even though the parameter changes smoothly.