Stability analysis of line patterns of an anisotropic interaction model

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 Kücken--Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.The work of the first author was partially supported by the EPSRC through grant EP/P031587/1. The work of the second author was supported by the Leverhulme Trust research project grant ``Novel discretizations for higher order nonlinear PDE"" (RPG-2015-69). The work of the third author was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German Academic Scholarship Foundation (Studienstiftung des Deutschen Volkes). The work of the fourth author was supported by the Leverhulme Trust project on breaking the non-convexity barrier, EPSRC grant EP/M00483X/1, the EPSRC Centre EP/N014588/1, the RISE projects CHiPS and NoMADS, the Cantab Capital Institute for the Mathematics of Information, and the Alan Turing Institute