Strategic Choice as Bifurcation: Amplitude–Phase Games on Stuart–Landau Networks

We introduce Amplitude–Phase Games, a class of continuous games in whicheach agent’s strategic state is a complex number zi = rieiθi , where ri encodesparticipation intensity and θi encodes relational orientation. Starting froma utility functional with participation benefit, saturation cost, and pairwisecoordination cost, we show that gradient-ascent adjustment dynamics yieldthe diffusively coupled Stuart–Landau equations. A normal-form argumentthen establishes that this result is generic: for any smooth strategic dynam-ics with rotational symmetry, Stuart–Landau is the canonical form near thebifurcation at which non-participation loses stability. Bifurcation analysis atthreescales—isolatedagent, two-agentgame, andmean-fieldlimit—identi-fies four macroscopic regimes (Active–Locked, Active–Unlocked, Quiescent–Locked, Exit), classified by two order parameters: aggregate participationRA and amplitude-weighted phase coherence Rθ. Both the Kuramoto mean-field game (forced participation, phase freedom only) and real-valued non-linear opinion dynamics (frozen phase, amplitude freedom only) are recov-ered as limiting cases, unifying three previously separate bridges betweengame theory and coupled-oscillator dynamics. Applied to Hirschman’s Exit–Voice–Loyalty framework, the regime map realizes these strategic categoriesasdynamicalattractors,withbifurcation-governedtransitionsthatreproduceempirically observed trajectories of withdrawal, co-optation, and staged de-parture.