Route to chaos and resonant triads interaction in a truncated Rotating Nonlinear shallow-water model

The route to chaos and phase dynamics in a rotating shallow-water model were rigorously examined using a five-mode Galerkin truncated system with complex variables. This system is valuable for investigating how large/meso-scales destabilize and evolve into chaos. Two distinct transitions into chaotic behaviour were identified as energy levels increased. The initial transition occurs through bifurcations following the Feigenbaum sequence. The subsequent transition, at higher energy levels, shows a shift from quasi-periodic states to chaotic regimes. The first chaotic state is mainly due to inertial forces governing nonlinear interactions. The second chaotic state arises from the increased significance of free surface elevation in the dynamics. A novel reformulation using phase and amplitude representations for each truncated variable revealed that phase components exhibit a temporal piece-wise locking behaviour, maintaining a constant value for a prolonged interval before an abrupt transition of $\pm\pi$, while amplitudes remain chaotic. It was observed that phase stability duration decreases with increased energy, leading to chaos in phase components at high energy levels. This is attributed to the nonlinear term in the equations, where phase components are introduced through linear combinations of triads with different modes. When locking durations vary across modes, the dynamics result in a stochastic interplay of multiple $\pi$ phase shifts, creating a stochastic dynamic within the coupled phase triads, observable even at minimal energy injections.