Oregonator — Belousov-Zhabotinsky Reaction

Oregonator: ε ẋ = qy − xy + x(1−x)
ε' ẏ = −qy − xy + fz
ż = x − z
X: HBrO₂, Y: Br⁻, Z: Ce⁴⁺ (oxidized catalyst). Oscillation period ~ minutes in experiment.

Limit cycle in 3D phase space. Projected onto X-Z plane. Hopf bifurcation occurs at critical parameter value — stable spiral → limit cycle. Limit cycle is a closed orbit, globally attracting.

As stoichiometric parameter f increases through f_c ≈ 1.0, stable fixed point loses stability via Hopf bifurcation → stable limit cycle with amplitude ~ √(f−f_c). Oscillation frequency at onset = Im(λ) of Jacobian eigenvalue.

Spiral waves in BZ reaction are rotating chemical waves. Core is a phase singularity. Curvature relation: c = c₀ + D·κ (Keener-Tyson). Spirals emerge from broken plane waves.