Canards in a bottleneck

TL;DR

This paper analyzes stationary solutions of a nonlinear Fokker-Planck equation with small diffusion in corridors with bottlenecks, revealing canard solutions and detailed bifurcation structures.

Contribution

It introduces a constructive geometric singular perturbation approach to classify solution profiles and identify canard solutions in bottleneck geometries.

Findings

Identified three main types of stationary profiles in bottleneck corridors.

Discovered canard solutions at the narrowest point of the bottleneck.

Provided a bifurcation diagram relating in- and outflow rates to solution types.

Abstract

In this paper we investigate the stationary profiles of a nonlinear Fokker-Planck equation with small diffusion and nonlinear in- and outflow boundary conditions. We consider corridors with a bottleneck whose width has a global nondegenerate minimum in the interior. In the small diffusion limit the profiles are obtained constructively by using methods from geometric singular perturbation theory (GSPT). We identify three main types of profiles corresponding to: (i) high density in the domain and a boundary layer at the entrance, (ii) low density in the domain and a boundary layer at the exit, and (iii) transitions from high density to low density inside the bottleneck with boundary layers at the entrance and exit. Interestingly, solutions of the last type involve canard solutions generated at the narrowest point of the bottleneck. We obtain a detailed bifurcation diagram of these solutions in terms of the in- and outflow rates. The analytic results based on GSPT are further corroborated by computational experiments investigating corridors with bottlenecks of variable width.