Supercritical Hopf Bifurcation

As μ crosses zero, a stable equilibrium loses stability and a stable limit cycle is born — amplitude grows as √μ, a soft onset of oscillation.

Phase plane (x, y)
Time series x(t)
Bifurcation diagram — amplitude vs μ
stableμ=-0.30limit cycle
1.0
0.50
—
Normal form: ṙ = μr − r³, θ̇ = ω (polar coordinates). Equivalently: ẋ = μx − ωy − (x²+y²)x, ẏ = ωx + μy − (x²+y²)y. Fixed point: origin (always exists). For μ<0: origin is stable spiral (eigenvalues μ±iω have negative real part). At μ=0: Hopf bifurcation — eigenvalues cross imaginary axis. For μ>0: origin becomes unstable spiral; stable limit cycle at r* = √μ. Amplitude scales as √(μ−μ_c) — the hallmark of a supercritical (soft) bifurcation. Found in: laser threshold, heart rhythm onset, circadian clocks, chemical oscillators.