Bifurcation of a Model for Adoption and Abandonment Dynamics

This study presents a comprehensive bifurcation-theoretic analysis of a nonlinear dynamical system that models the adoption and abandonment of emerging technologies. By extending previous frameworks for user adoption dynamics, the model incorporates both endogenous and exogenous abandonment mechanisms. We examine critical transitions in user behavior, including saddle-node, Hopf, Bautin, and Bogdanov–Takens bifurcations, as well as cusp singularities that lead to hysteresis and multistability. Our results provide a detailed map of parameter regimes that govern the transition from dominance to decline of digital platforms, offering valuable insights for forecasting user engagement in a rapidly evolving technological ecosystem.