From heteroclinic loops to homoclinic snaking in reversible systems: rigorous forcing through computer-assisted proofs
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912497355718656 |
|---|---|
| author | Berg, Jan Bouwe van den Duchesne, Gabriel William Lessard, Jean-Philippe |
| author_facet | Berg, Jan Bouwe van den Duchesne, Gabriel William Lessard, Jean-Philippe |
| contents | Homoclinic snaking is a widespread phenomenon observed in many pattern-forming systems. Demonstrating its occurrence in non-perturbative regimes has proven difficult, although a forcing theory has been developed based on the identification of patterned front solutions. These heteroclinic solutions are themselves challenging to analyze due to the nonlinear nature of the problem. In this paper, we use computer-assisted proofs to find parameterized loops of heteroclinic connections between equilibria and periodic orbits in time reversible systems. This leads to a proof of homoclinic snaking in both the Swift-Hohenberg and Gray-Scott problems. Our results demonstrate that computer-assisted proofs of continuous families of connecting orbits in nonlinear dynamical systems are a powerful tool for understanding global dynamics and their dependence on parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16798 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From heteroclinic loops to homoclinic snaking in reversible systems: rigorous forcing through computer-assisted proofs Berg, Jan Bouwe van den Duchesne, Gabriel William Lessard, Jean-Philippe Dynamical Systems Numerical Analysis Analysis of PDEs 34C37 (Primary) 37M21, 35B36, 65G40, 65T40, 42A10, 37C79 (Secondary) Homoclinic snaking is a widespread phenomenon observed in many pattern-forming systems. Demonstrating its occurrence in non-perturbative regimes has proven difficult, although a forcing theory has been developed based on the identification of patterned front solutions. These heteroclinic solutions are themselves challenging to analyze due to the nonlinear nature of the problem. In this paper, we use computer-assisted proofs to find parameterized loops of heteroclinic connections between equilibria and periodic orbits in time reversible systems. This leads to a proof of homoclinic snaking in both the Swift-Hohenberg and Gray-Scott problems. Our results demonstrate that computer-assisted proofs of continuous families of connecting orbits in nonlinear dynamical systems are a powerful tool for understanding global dynamics and their dependence on parameters. |
| title | From heteroclinic loops to homoclinic snaking in reversible systems: rigorous forcing through computer-assisted proofs |
| topic | Dynamical Systems Numerical Analysis Analysis of PDEs 34C37 (Primary) 37M21, 35B36, 65G40, 65T40, 42A10, 37C79 (Secondary) |
| url | https://arxiv.org/abs/2507.16798 |