HOPF BIFURCATION — NORMAL FORM
PARAMETERS
μ (bifurcation param):
0.00
Initial radius r₀:
0.20
Trail length:
200
REGIME
CRITICAL (μ=0)
LIMIT CYCLE RADIUS
—
EIGENVALUES λ₁,₂
μ ± i
Normal Form:
ẋ = μx − y − x(x²+y²)
ẏ = x + μy − y(x²+y²)
In polar: ṙ = r(μ − r²), θ̇ = 1
• μ < 0: stable fixed point, trajectories spiral in
• μ = 0: weakly stable (nonlinear), Hopf point
• μ > 0: unstable fixed point + stable limit cycle of radius √μ
This is the
supercritical
Hopf bifurcation — the prototypical birth of oscillation.