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On the stability of Rotating States in Second-Order Self-Propelled Multi-Particle Systems

Carl Kolon, Constantine Medynets, Irina Popovici

TL;DR

This work analyzes the long-time behavior of a planar system of $n$ identical self-propelled particles with second-order dynamics and quadratic, rotationally symmetric coupling. By introducing rotating-frame coordinates and exploiting center-manifold techniques tailored to non-isolated fixed points, the authors prove that all rotating states are stable and that nearby trajectories converge to nearby rotating states, with exponential rates for odd $n$ and exponential or $1/\sqrt t$ rates for even $n$. The analysis handles both non-degenerate and degenerate ring states, and exposes limitations of standard Taylor-based center-manifold approximations, motivating a novel approximation scheme anchored by a continuum of equilibrium points. These results rigorously characterize the asymptotic ring-like patterns (mill/ring states) observed in swarming models and establish a principled framework for the convergence to coherent rotational patterns in second-order swarm systems.

Abstract

In this paper, we study the dynamics of a system of $n$ coupled, self-propelled particles: $\ddot r_k = (α-β|\dot r_k|^2)\dot r_k - \fracγ{n}\sum_{m=1}^n(r_k-r_m)$, $r_k\in \mathbb R^2.$ Numerical experiments indicate that, for a large set of initial conditions, after an initial drift, the center of mass converges to a stationary point, with each particle eventually rotating around it with constant angular velocity. The distribution of particles on the circle need not be uniform. These limit configurations, where all particles rotate in the same direction, are termed {\it rotating states} . We prove that rotating states are stable and that every solution that starts sufficiently close, asymptotically approaches a rotating state, exponentially fast if $n$ is odd, or at a rate that may be exponential or $\frac{1}{\sqrt t} $ if $n$ is even. The proof uses a new approximation technique for the flow on the center manifold in the presence of non-isolated fixed points.

On the stability of Rotating States in Second-Order Self-Propelled Multi-Particle Systems

TL;DR

This work analyzes the long-time behavior of a planar system of identical self-propelled particles with second-order dynamics and quadratic, rotationally symmetric coupling. By introducing rotating-frame coordinates and exploiting center-manifold techniques tailored to non-isolated fixed points, the authors prove that all rotating states are stable and that nearby trajectories converge to nearby rotating states, with exponential rates for odd and exponential or rates for even . The analysis handles both non-degenerate and degenerate ring states, and exposes limitations of standard Taylor-based center-manifold approximations, motivating a novel approximation scheme anchored by a continuum of equilibrium points. These results rigorously characterize the asymptotic ring-like patterns (mill/ring states) observed in swarming models and establish a principled framework for the convergence to coherent rotational patterns in second-order swarm systems.

Abstract

In this paper, we study the dynamics of a system of coupled, self-propelled particles: , Numerical experiments indicate that, for a large set of initial conditions, after an initial drift, the center of mass converges to a stationary point, with each particle eventually rotating around it with constant angular velocity. The distribution of particles on the circle need not be uniform. These limit configurations, where all particles rotate in the same direction, are termed {\it rotating states} . We prove that rotating states are stable and that every solution that starts sufficiently close, asymptotically approaches a rotating state, exponentially fast if is odd, or at a rate that may be exponential or if is even. The proof uses a new approximation technique for the flow on the center manifold in the presence of non-isolated fixed points.

Paper Structure

This paper contains 4 sections, 5 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Configurations with stationary center of mass.
  3. A brief outline of the paper.
  4. Change of Coordinates about a Given Rotating State

Key Result

Theorem 1.2

Every rotating state solution of Equation (eqnMainModel) is stable: for any $\epsilon >0$ there exists $\delta >0$ such that given any initial center of mass location $\mathrm R_0$ and polar angles $\theta_{0,k}$ with $\sum _{k=1}^n e^{i\theta_{0,k}}=0$, if a solution $\{\mathrm r_k(t)\}$ of (eqnMai Furthermore, every solution that starts near a rotating state converges to a nearby rotating state:

Figures (2)

  • Figure 1: Limit patterns for (1), with $\mathrm A=1, n=3$ (left) and $n=10$ (right). Left: All agents and the center of mass exhibit sustained oscillations. Agent positions at $t = 0$ (hollow dots) and at $t = 50$ (solid dots) are shown, with the center of mass marked by a red dot. Right: The center of mass $R(t)$ (dashed) becomes nearly stationary. Agents stabilize into uniform circular motion centered at $R(t)$ with relative phases ${\theta_k}$ satisfying $\sum_{k=1}^{n} e^{i\theta_k} = 0$.
  • Figure 2: An illustration of a 7-agent, 28-dimensional system partitioned into a set of three 1-dimensional Van der Pol oscillators (the square markers) and four rotating agents (the hollow markers), having $R(t)=(0,0)$ for all $t$. Left: Agent trajectories from $t = 0$ to $t = 1.5$ are shown as solid curves; their corresponding limit cycles are indicated by dashed curves. Right: projection of the curve $(x_1(t), x_2(t), \dots \dot y_7(t))\subset \mathbb R^{28}$ onto the 3-dimensional subspace $(x_1, x_4, \dot x_1).$

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 1 more