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

TL;DR

This paper explores how phase elements on the unit circle aggregate under parametric external fields with oscillating attraction strength and rotation, revealing complex nonlinear behaviors and stability structures.

Contribution

It introduces a model for nonlinear phase aggregation under time-varying external fields and uncovers Arnold tongue-like structures and rich dynamical regimes through analysis and simulations.

Findings

Identification of Arnold tongue-like structures in parameter space

Complete aggregation within wedge-shaped stability regions

Rich nonlinear behaviors including attractors and quasi-periodic trajectories

Abstract

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 ${d\alpha/dt} = \lambda(t)\sin 2(\alpha - \phi(t))$ with $\lambda(t) = \cos(\omega_1 t)$ and $\phi(t) = \omega_2 t$, 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 $(\omega_1, \omega_2)$, 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.