1. Large Fronts in Nonlocally Coupled Systems Using Conley–Floer Homology.
- Author
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Bakker, Bente Hilde and van den Berg, Jan Bouwe
- Subjects
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VECTOR fields , *PHASE space , *DYNAMICAL systems , *MORSE theory , *MATHEMATICAL convolutions - Abstract
In this paper, we study travelling front solutions for nonlocal equations of the type ∂ t u = N ∗ S (u) + ∇ F (u) , u (t , x) ∈ R d. Here, N ∗ denotes a convolution-type operator in the spatial variable x ∈ R , either continuous or discrete. We develop a Morse-type theory, the Conley–Floer homology, which captures travelling front solutions in a topologically robust manner, by encoding fronts in the boundary operator of a chain complex. The equations describing the travelling fronts involve both forward and backward delay terms, possibly of infinite range. Consequently, these equations lack a natural phase space, so that classic dynamical systems tools are not at our disposal. We therefore develop, from scratch, a general transversality theory, and a classification of bounded solutions, in the absence of a phase space. In various cases the resulting Conley–Floer homology can be interpreted as a homological Conley index for multivalued vector fields. Using the Conley–Floer homology, we derive existence and multiplicity results on travelling front solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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