Stability analysis of line patterns of an anisotropic interaction model

Jose A. Carrillo, Bertram During, Lisa Maria Kreusser, Carola Bibiane Schonlieb

Research output: Contribution to journalArticlepeer-review

4 Citations (SciVal)

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.

Original languageEnglish
Pages (from-to)1798-1845
Number of pages48
JournalSIAM Journal on Applied Dynamical Systems
Volume18
Issue number4
DOIs
Publication statusPublished - 15 Oct 2019

Keywords

  • Aggregation
  • Dynamical systems
  • Pattern formation
  • Swarming

ASJC Scopus subject areas

  • Analysis
  • Modelling and Simulation

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