Stability analysis of line patterns of an anisotropic interaction model

TL;DR

This paper analyzes the stability of line patterns in an anisotropic particle interaction model inspired by fingerprint formation, revealing conditions for stable straight lines and their relation to anisotropic forces.

Contribution

It introduces a stability analysis of straight line patterns in an anisotropic interaction model, extending understanding of pattern formation with tensor-dependent forces.

Findings

Stable straight vertical lines exist for large particle numbers.

Stability depends on force decay rates and anisotropy.

Results relate to fingerprint pattern simulation models.

Abstract

Motivated by the formation of fingerprint patterns we consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In addition, the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. Central to this pattern formation are straight line patterns. For a given spatially homogeneous tensor field, we show that there exists a preferred direction of straight lines, i.e.\ straight vertical lines can be stable for sufficiently many particles, while many other rotations of the straight lines are unstable steady states, both for a sufficiently large number of particles and in the continuum limit. For straight vertical lines we consider specific force coefficients for the stability analysis of steady states, show that stability can be achieved for exponentially decaying force coefficients for a sufficiently large number of particles and relate these results to the K\"ucken-Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.