Dual-moon forced dynamics and nonlinear aggregation in Saturn's F ring: From quasi-periodicity to modulated oscillations

We develop a minimal nonlinear model to investigate the oscillatory dynamics of Saturn’s F ring under dual-moon forcing from Prometheus and Pandora. The model extends classical predator–prey dynamics by incorporating both a nonlinear mass aggregation term $k{M}^n$ and explicit dual-frequency forcing, capturing how higher-order coagulation physics interacts with multi-moon perturbations. Through extensive numerical integration and dynamical systems analysis, including time-series, spectral, stroboscopic mapping, and rotation number diagnostics, we identify distinct dynamical regimes controlled by the parameters $n$ and $k$. For moderate nonlinearity $(n=1.28,k=0.54)$, the system exhibits regular quasi-periodic motion on a two-torus, characterized by smooth amplitude modulation and discrete spectral lines. As nonlinearity increases $(n=1.30,k=0.62)$, the dynamics transition to strongly modulated oscillations with intermittent phase slips, broadened Poincaré bands, and sideband-rich spectra. A rotation number heatmap reveals organized structures in parameter space, with smooth quasi-periodic regions bounded by near-locking bands analogous to Arnold tongues. Our results demonstrate that the F ring’s complex morphology can emerge from deterministic multi-frequency dynamics rather than stochastic processes, with the system operating near critical boundaries where small parameter variations can trigger macroscopic reorganization. The model provides a framework for understanding pattern formation in other driven granular systems while offering testable predictions for ring observations.