Canards in a bottleneck
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910113907867648 |
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| author | Iuorio, Annalisa Jankowiak, Gaspard Szmolyan, Peter Wolfram, Marie-Therese |
| author_facet | Iuorio, Annalisa Jankowiak, Gaspard Szmolyan, Peter Wolfram, Marie-Therese |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16652 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Canards in a bottleneck Iuorio, Annalisa Jankowiak, Gaspard Szmolyan, Peter Wolfram, Marie-Therese Dynamical Systems Analysis of PDEs 34B16, 34E13, 34E15 (Primary), 35B25, 35B40, 65L11, 76A30 (Secondary) 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. |
| title | Canards in a bottleneck |
| topic | Dynamical Systems Analysis of PDEs 34B16, 34E13, 34E15 (Primary), 35B25, 35B40, 65L11, 76A30 (Secondary) |
| url | https://arxiv.org/abs/2211.16652 |