标题： 相位元件在参数外部场作用下在单位圆上的非线性聚集 显示英文标题

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

摘要： 我们研究在参数调制外部场作用下，分布在单位圆上的相位元件的非线性聚集动力学。 我们的模型受流体中鳞片粒子旋转的启发，采用方程${d\alpha/dt} = \lambda(t)\sin 2(\alpha - \phi(t))$，其中$\lambda(t) = \cos(\omega_1 t)$和$\phi(t) = \omega_2 t$表示一种切换旋转吸引装置，其中吸引强度振荡，而吸引点以独立频率旋转。 通过数值模拟和分析方法，我们在参数空间$(\omega_1, \omega_2)$中发现类似阿诺德舌结构，在这里初始各向同性的相位分布聚集为高度各向异性的状态。 完全聚集发生在从分岔点辐射出的楔形稳定区域内，形成具有特征斜率关系的带状结构。 动力学表现出丰富的非线性行为，包括在由聚集度（$I$）、场对齐度量（$O$）和时间变化（$P$）所张成的简化指标空间中的吸引子、极限环和准周期轨迹。 我们的发现揭示了在竞争时间调制下集体相位动力学的基本原理，其潜在应用涵盖从生物同步到社会经济动力学以及可控制的集体系统。 显示英文摘要

摘要： 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.