Hopf Bifurcation Normal Form

The supercritical Hopf bifurcation in polar coordinates: the radial equation ṙ = r(μ − r²) shows a stable fixed point at r=0 for μ<0, and a stable limit cycle at r=√μ for μ>0. The bifurcation at μ=0 creates oscillations from nothing.

ṙ = r(μ − r²)  |  θ̇ = ω  |  Limit cycle: r* = √μ  |  Period: T = 2π/ω