Hopf Bifurcation & Amplitude Death

Two coupled Stuart–Landau oscillators: as coupling κ increases, oscillations die (amplitude death) when the oscillators pull each other to equilibrium

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Stuart–Landau equation: ż = (μ + iω)z − |z|²z + κ(w−z), where z,w ∈ ℂ are the two oscillators. For μ<0: stable fixed point (no oscillation). At μ=0: supercritical Hopf bifurcation — a stable limit cycle appears with amplitude √μ. Amplitude death occurs when coupling κ is strong enough and frequencies differ (Δω≠0): oscillators suppress each other and both collapse to zero, even though each would oscillate in isolation.