Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.07992

Fragmentation and Cluster Prediction in One-Dimensional Finite-Range Normalized Alignment Dynamics

Jiangning Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study fragmentation in a one-dimensional finite-range normalized alignment system, where each agent relaxes its velocity toward the average velocity of agents within a fixed interaction radius. Because the communication graph depends on the agents' positions, edges can disappear as agents separate, leading to multiple asymptotic velocity clusters. We first consider spatially ordered initial data with velocities ordered in the same direction. In this expansive regime, we prove that velocity ordering is forward invariant, all pairwise separations are nondecreasing, and communication edges can only be deleted. Hence, the dynamics undergo finitely many topology changes. Between changes, the velocity dynamics form a linear system generated by a random-walk Laplacian. Using its group inverse, we derive an exact terminal-separation formula for every active edge and obtain a finite recursive procedure for determining the terminal communication graph and the asymptotic velocity of each cluster. For path graphs, this theory becomes explicit: a tridiagonal Green kernel gives a necessary-and-sufficient criterion for finite-time fragmentation and a sharp critical alignment strength separating fragmentation from mono-flocking. We also identify a class of non-monotone initial velocities that enters the expansive regime before the first topology change and derive Gaussian statistics for terminal separations under random initial velocities. For this class of ordered and safely entering initial data, our results provide a rigorous mechanism for the spontaneous group division observed in earlier numerical studies. The general switching problem for arbitrary initial configurations remains open. Reproducible computations validate the path threshold, first-fragmentation-time prediction, and finite-event recursion against independent direct integration of the switching dynamics.

Explore related subjects

math.DSnlin.AO

Keep this discovery

BibTeXRIS

Jiangning Chen. 2026-09-07. Fragmentation and Cluster Prediction in One-Dimensional Finite-Range Normalized Alignment Dynamics. https://arxiv.org/abs/2609.07992

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗