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. We show that there exists a preferred direction of straight lines, i.e. straight vertical lines can be stable for sufficiently many particles, while 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.AMSC: 35B36, 35Q92, 70F10, 70F45, 82C22

Analysis of Anisotropic Interaction Equations for Fingerprint Simulations

“…For this choice of the tensor field we expect straight vertical lines as steady states which is also suggested by the numerically computed steady states for the particle model (1) and its continuum counterpart (3) in Figure 1, and the numerical results in [4]. In fact, we showed rigorously in [6] rigorously that particles distributed equidistantly along straight vertical lines are stable steady states to the particle model (1) for exponentially decaying force coefficients f A and f R for any particle number N ∈ N sufficiently large, while all other rotations of straight lines are unstable steady states. This stability result is important for understanding the robustness of patterns in applications such as, for instance, for fingerprint simulations.…”

“…This stability result is important for understanding the robustness of patterns in applications such as, for instance, for fingerprint simulations. Based on the theoretical results in [4,6] we simulate fingerprints with variable ridge distances as steady states to the particle model (1) in [7]. However, as the number of particles N tends to infinity, the particle simulations of (1) become very inefficient.…”

Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics

“…A special instance of this model has been introduced in [28] for simulating fingerprint patterns where the authors assumed a specific form of the force F . The particle model in its general form (9) has been studied in [9,17,20]. The existence of different kinds of steady states, including steady states in the form of lines, is investigated in [9], both for the particle model (9) and its continuum counterpart (1).…”

“…The stationary solutions to (1) can be regarded as solutions with one-dimensional support [9] and may be constant on its support. The direction of the line patterns depends on the choice of the tensor field T with its vector fields s and l. For purely repulsive forces along s and short-range repulsive, long-range attractive forces along l, the stability of line patterns is proven for spatially homogeneous tensor fields in [17], based on a stability analysis of (9). The proof considers perturbations of equidistantly distributed particles along lines and shows that line patterns along s are stable, while most other rotations including line patterns along l are unstable.…”

An anisotropic interaction model with collision avoidance

“…Moreover, it is straight forward to include interaction with obstacles, which can be represented at fixed positions with artificial velocities. The approach is inspired by [11,16,30], where the formation of fingerprint patterns is studied with the help of an anisotropic force field. Further studies may consider the anisotropy parameter to be density or obstacle dependent.…”