Threshold-Induced Bifurcation and Chaos in a One-Dimensional Coastal Risk Adaptation Map

Coastal disaster risk is often shaped by threshold mechanisms, where mitigation actions are activated after residual risk exceeds a critical level. This study proposes a one-dimensional threshold adaptation map for describing the evolution of normalized residual coastal risk under hazard amplification and adaptive response. The model combines logistic-type risk growth with a smooth sigmoidal threshold function representing the gradual activation of mitigation measures. Analytical results are obtained for the invariant interval, fixed points, local stability, and bifurcation conditions. The analysis shows that the sharpness of the threshold response plays a central role in determining the stability of the positive equilibrium. Numerical simulations using bifurcation diagrams, cobweb plots, time series, and Lyapunov exponents reveal period-doubling cascades and chaotic dynamics when the hazard amplification or threshold sharpness increases. These results indicate that adaptation may reduce residual risk when implemented gradually, but excessively abrupt threshold responses may generate persistent oscillations and chaotic fluctuations. The proposed model provides a simple mathematical framework for understanding threshold-induced instability in coastal disaster risk dynamics and highlights the importance of designing adaptive interventions that are dynamically smooth as well as effective.