Normal form: ṙ = μr − r³ (supercritical) vs. ṙ = μr + r³ − r⁵ (subcritical)
Parameters
Type
Supercritical Hopf
ṙ = μr − r³ Stable limit cycle born at μ=0, amplitude ~ √μ. No hysteresis.
Current State
μ-0.50
Fixed point r=0stable
Limit cycle r*none
Amplitude ~ √μ–
Physics
The Hopf bifurcation is the birth of oscillation from a fixed point. In polar coordinates:
Supercritical (ṙ=μr−r³): stable limit cycle appears continuously at μ=0; amplitude → 0 as μ→0⁺.
Subcritical (ṙ=μr+r³−r⁵): unstable limit cycle exists for μ<0, then merges with stable one at μ=0; system jumps discontinuously — bistability & hysteresis.