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

TL;DR

This work analyzes nonlinear phase aggregation on the unit circle for phase elements driven by two independent periodic external modulations, encoded by $\dot{\alpha_i}=\lambda(t)\sin 2(\alpha_i-\phi(t))$ with $\lambda(t)=\cos(\omega_1 t)$ and $\phi(t)=\omega_2 t$. Through numerical simulations and targeted analytical reductions, the authors reveal Arnold tongue–like resonance bands and wedge-shaped regions in $(\omega_1,\omega_2)$ that drive complete aggregation, and they show how Adler-type reductions and specific resonance cases ($\omega_1=2\omega_2$ and $\omega_1=\omega_2$) underpin these structures. They introduce an extended reduced space $\mathbb{A}=(I,O,P)$ to capture history-dependent responses and demonstrate invariance of the dynamics under rotational frame changes, extending the framework to rotational flows. The findings connect flow-visualization phenomena with nonlinear synchronization theory, offering a unified view of resonance locking, and suggesting practical routes to control and infer collective dynamics in fluids and related systems. The work thus provides theoretical and numerical tools to predict and analyze aggregation patterns under complex temporal modulations with potential applications ranging from biological synchronization to socio-economic dynamics.

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 ${dα/dt} = λ(t)\sin 2(α- φ(t))$ 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.

Figures (5)

• Figure 1: Time evolution of $N=100$ phase elements initially distributed isotropically at $t=0$. Left panel (Case A) shows $(\omega_1, \omega_2) = (1.1, 1.1)$ where phases $\alpha$ converge to an anisotropic state over time. Right panel (Case B) shows $(\omega_1, \omega_2) = (1.1, 1.2)$ where the system periodically alternates between isotropic and anisotropic states.
• Figure 2: Contour map of $\langle I \rangle_{[T/2, T]}$ for $N=20$. Horizontal and vertical axes are $\omega_1$ and $\omega_2$, respectively. Parameter resolution is $\Delta\omega = 2.0\times{10^{-3}}$. Solid and dashed red lines indicate $\omega_1 = \omega_2$ and $\omega_1 = 2\omega_2$, respectively. In the bright region where $\langle I \rangle_{[T/2, T]} \approx 1$, all phase elements completely aggregate, forming contiguous wedge-shaped regions that originate from (0,1), with a prominent band having slope $1/2$. The red solid line $\omega_1 = \omega_2$ intersects the boundary of the bright region at $\omega_1\approx 1.16$. Points A and B correspond to Fig. \ref{['fig:fig-0']}A and B, respectively.
• Figure 3: Representative slices of $\langle I \rangle_{[T/2,T]}$ for $N=20$, illustrating the detailed behavior near the critical regions. Case (a) ($\omega_2 = 1$): Multiple plateau regions with $\langle I \rangle \approx 1$ are observed, each followed by sharp departures near the corresponding critical points. Case (b) ($\omega_1 = 1/2$): Similarly, multiple plateau regions are present, each followed by sharp departures near the corresponding critical points. These slices were computed with slightly different parameters ($\Delta\omega=2.0\times 10^{-4}$, $T=50 \cdot 2\pi / \omega_1$) for higher resolution.
• Figure 4: Contour plot of ${\langle I \rangle_{[T/2,T]}}$ in the $\omega_2$–$\epsilon$ plane for $N=20$, illustrating how the aggregation regions depend on the $\omega_1$. Case (a) ($\omega_1=0$): A single tongue-shaped region with ${\langle I \rangle_{[T/2,T]} \approx 1}$ appears in the domain $\epsilon > \omega_2$. Case (b) ($\omega_1=1/2$): Multiple tongue-shaped regions with ${\langle I \rangle_{[T/2,T]} \approx 1}$ emerge.
• Figure 5: Examples of $(I,O,P)$ trajectories for various $\omega_1$, $\omega_2$ settings, starting from isotropic initial orientations of $\alpha_i$ for $N=5$,$000$. (A) $\omega_1 = \omega_2 = 0$: Convergence to attractor $(1,1,0)$ under static external field. (B) $\omega_1 \neq 0$, $\omega_2 = 0$: Periodic motion with $O$-$P$ describing closed orbit. (C) $\omega_1 = 0$, $\omega_2 \neq 0$: Elliptical periodic orbit. (D) $\omega_2/\omega_1 = 1/2$: Limit cycle in complete aggregation state. (E,F) Complex quasi-periodic trajectories by superposition of basic orbits.