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

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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.

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Authors and Affiliations

  1. Department of Mathematics, VU Amsterdam, De Boelelaan 1081, 1081 HV, Amsterdam, The Netherlands

    Jan Bouwe van den Berg & Rob Van der Vorst

  2. Department of Mathematics, Rutgers University, The State University of New Jersey, Piscataway, NJ, 08854, USA

    Marcio Gameiro

  3. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, São Carlos, SP, 13560-970, Brazil

    Marcio Gameiro

  4. Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada

    Jean-Philippe Lessard

Authors
  1. Jan Bouwe van den Berg
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  2. Marcio Gameiro
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  3. Jean-Philippe Lessard
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  4. Rob Van der Vorst
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Correspondence to Jean-Philippe Lessard.

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Communicated by Peter Bubenik.

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van den Berg, J.B., Gameiro, M., Lessard, JP. et al. Toward Computational Morse–Floer Homology: Forcing Results for Connecting Orbits by Computing Relative Indices of Critical Points. Found Comput Math (2023). https://doi.org/10.1007/s10208-023-09623-w

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  • DOI: https://doi.org/10.1007/s10208-023-09623-w

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