Stability analysis of line patterns Of an anisotropic interaction model

Carrilo, José A, Düring, Bertram, Kreusser, Lisa Maria and Schöenlieb, Carola-Bibiane (2019) Stability analysis of line patterns Of an anisotropic interaction model. SIAM Journal on Applied Dynamical Systems. ISSN 1536-0040 (Accepted)

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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 Kuecken-Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.

Item Type: Article
Keywords: Aggregation, swarming, pattern formation, dynamical systems
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Numerical Analysis and Scientific Computing Research Group
Subjects: Q Science > QA Mathematics
Depositing User: Amelia Redman
Date Deposited: 21 Aug 2019 12:43
Last Modified: 21 Aug 2019 13:19
URI: http://sro.sussex.ac.uk/id/eprint/85577

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Project NameSussex Project NumberFunderFunder Ref
Novel discretisations of higher-order nonlinear PDEG1603LEVERHULME TRUSTRPG-2015-069