Tipping Points & Bifurcations
Catastrophic regime shifts: saddle-node, pitchfork, and Hopf bifurcations
System
Saddle-Node
Pitchfork
Cusp Catastrophe
dx/dt = r + x² (fold/saddle-node)
Fixed pts: x* = ±√(-r)
Tipping point at r = 0: saddle-node bifurcation
Controls
Control param r
0.00
Bias c
0.00
Initial state x₀
0.50
⏸ Pause
Reset
State
r
—
Current x
—
dx/dt
—
Regime
—
Early warning signals:
• Critical slowing down (λ→0)
• Rising variance & autocorrelation
• Flickering near threshold