Who Holds the Parameters? A Stuart–Landau Account of Smooth and Sudden Transitions in Human Collectives

Why are some collective transitions smooth and others sudden? Modelling collective coordination with the Stuart–Landau equation — the Hopf normal form near the instability at which coordinated order forms or fails — we show that the answer lies in the first Lyapunov coefficient, whose sign determines whether the transition is supercritical (continuous, reversible) or subcritical (abrupt, hysteretic). This coefficient is the dynamical origin of the splitting factor in catastrophe models of organisational behaviour. The Bautin bifurcation, where the coefficient vanishes, is the boundary between regimes. We propose task interdependence, deadline pressure, group size, reward-structure thresholds, and boundary permeability as candidates for the second parameter, and state the predictions as propositions. Within the same framework, external influence enters as forcing (instruction on the state) or parameter-setting (constraint on the landscape), and autonomy becomes parameter ownership. The collapse modes of coupled oscillator systems map onto organisational pathologies of exit, disengagement, and fragmentation, realising Hirschman's exit–voice–loyalty triad as four quadrants of the amplitude–phase plane. Formation, collapse, and division are three fates of collective order within one structure.