Delay-induced bifurcations and hyper-chaotic behavior in shallow-water wave dynamics

Shallow-water wave equations provide a fundamental framework for describing wave propagation in coastal and ocean environments, including tsunamis, storm surges, and long-wave dynamics. In practical marine systems, wave evolution is often influenced by memory effects arising from delayed physical interactions, yet the impact of these effects on stability and predictability remains poorly understood. This study investigates the dynamical consequences of local and non-local distributed delays in a delayed shallow-water wave equation. Using traveling-wave reduction, center manifold theory, and normal form analysis, the governing partial differential equation is transformed into finite-dimensional dynamical systems whose equilibrium structure and bifurcation properties are rigorously characterized. The analysis reveals that memory effects fundamentally alter wave dynamics by inducing fold, Hopf, cusp, and Fold-Hopf bifurcations, leading to transitions among steady, oscillatory, and unstable wave states. Numerical investigations further demonstrate hyper-chaotic dynamics under both local and non-local delay mechanisms, confirmed through Lyapunov exponents, bifurcation diagrams, and Poincaré sections. The results show that distributed memory acts as a key mechanism governing wave instability and long-term unpredictability. These findings provide new insights into delayed wave processes relevant to coastal engineering, wave forecasting, ocean monitoring, and environmental fluid dynamics.