Nonlinear Aggregation of Phase Elements on the Unit Circle under Parametric External Fields

We investigate nonlinear aggregation dynamics of phase elements distributed on the unit circle under parametrically modulated external fields. Our model, inspired by flaky particle rotation in fluids, employs the equation ${dα/dt} = λ(t)\sin 2(α- ϕ(t))$ with $λ(t) = \cos(ω_1 t)$ and $ϕ(t) = ω_2 t$, representing a switching rotating attractive device where the attractive strength oscillates while the attractive point rotates at independent frequencies. Through numerical simulations and analytical approaches, we discover Arnold tongue-like structures in parameter space $(ω_1, ω_2)$, where initially isotropic phase distributions aggregate into highly anisotropic states. Complete aggregation occurs within wedge-shaped stability regions radiating from bifurcation points, forming band structures with characteristic slope relationships. The dynamics exhibit rich nonlinear behavior including attractors, limit cycles, and quasi-periodic trajectories in reduced indicator space spanned by aggregation degree ($I$), field-alignment measure ($O$), and temporal variation ($P$). Our findings reveal fundamental principles governing collective phase dynamics under competing temporal modulations, with potential applications spanning from biological synchronization to socio-economic dynamics and controllable collective systems.