An oscillation born the instant a resting state loses its stability. As one parameter crosses a critical value, a stable fixed point turns unstable and a self-sustained limit cycle is born — the Jacobian's complex-conjugate eigenvalues cross the imaginary axis. Rendered, not quoted: the eigenvalues, the fixed point, and the emergent cycle are computed live below.
source E. Hopf, Abzweigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems, Ber. Math.-Phys. Kl. Sachs. Akad. Wiss. Leipzig 94 (1942) 1–22 — no stable open link (AMBER); English translation in Marsden & McCracken, The Hopf Bifurcation and Its Applications (1976). Also Andronov–Hopf, Poincare 1892.
The normal form on the plane, near the fixed point at the origin:
x' = μx − ωy − (x²+y²)x
y' = ωx + μy − (x²+y²)y
Its Jacobian at the origin is [[μ,−ω],[ω,μ]], eigenvalues μ ± iω. The real part is μ; it passes through zero as μ increases. Supercritical: a stable cycle of radius √μ appears for μ>0.
Assumption (AMBER): the supercritical normal form (first Lyapunov coefficient < 0). A subcritical Hopf instead throws off an unstable cycle and can jump. Stated, not hidden.
An oscillation out of stillness. The moment the eigenvalues cross into instability is where the-limit-cycle gets its birth — one route the-bifurcation takes. This sphere is that crossing, watched frame by frame.
Re-checks the two regimes live: real part negative below the critical value, positive above (a genuine crossing), and a cycle born only above. Tamper window 6 and this badge flips.
Control parameter μ (drag it across the critical value 0). Rotation ω = 1.0 fixed. A seeded ring of starting points, integrated by RK4 at dt = 0.01.
At μ=0.30, ω=1.0: eigenvalues 0.300 ± 1.000 i (unstable spiral), stable limit cycle at radius √0.30 = 0.5477 — the oscillation is born.
Crossing the imaginary axis is necessary, not sufficient. Without transversality (eigenvalues cross with nonzero speed) and a nonzero first Lyapunov coefficient, the theorem gives no cycle. A degenerate crossing can produce a center, a slow drift, or nothing.
And this is the planar normal form. Real high-dimensional systems reduce to it only on a 2D center manifold — a reduction that must be earned, not assumed.
Planted void (disclosed): force the eigenvalues' real part to stay negative past the critical value — no crossing, so no cycle is ever born. The witness in window 7 catches it live.