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Foundations of Computational Mathematics
Erschienen in:

17.10.2023

Toward Computational Morse–Floer Homology: Forcing Results for Connecting Orbits by Computing Relative Indices of Critical Points

verfasst von: Jan Bouwe van den Berg, Marcio Gameiro, Jean-Philippe Lessard, Rob Van der Vorst

Erschienen in: Foundations of Computational Mathematics | Ausgabe 5/2024

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Abstract

To make progress toward better computability of Morse–Floer homology and thus enhance the applicability of Floer theory, it is essential to have tools to determine the relative index of equilibria. Since even the existence of nontrivial stationary points is often difficult to accomplish, extracting their index information is usually out of reach. In this paper, we establish a computer-assisted proof approach to determining relative indices of stationary states. We introduce the general framework and then focus on three example problems described by partial differential equations to show how these ideas work in practice. Based on a rigorous implementation, with accompanying code made available, we determine the relative indices of many stationary points. Moreover, we show how forcing results can be then used to prove theorems about connecting orbits and traveling waves in partial differential equations.
Vorheriger Artikel A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow

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Metadaten
Titel
Toward Computational Morse–Floer Homology: Forcing Results for Connecting Orbits by Computing Relative Indices of Critical Points
verfasst von
Jan Bouwe van den Berg
Marcio Gameiro
Jean-Philippe Lessard
Rob Van der Vorst
Publikationsdatum
17.10.2023
Verlag
Springer US
Erschienen in
Foundations of Computational Mathematics / Ausgabe 5/2024
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-023-09623-w

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