Brusselator Limit Cycle

Chemical oscillator: X,Y concentrations spiral toward a stable limit cycle — Hopf bifurcation as B crosses 1+A²

B=2.5 > 1+A²=2.0 → LIMIT CYCLE
Fixed point: X*=1.00, Y*=2.50
Cycle amplitude ≈ —
Brusselator: dX/dt = A − (B+1)X + X²Y, dY/dt = BX − X²Y.
Fixed point (A, B/A). Stability: B < 1+A² → stable spiral. B > 1+A² → Hopf bifurcation → limit cycle.