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Register a new userFloer Homology: From Generalized Morse-Smale Dynamical Systems to Formans Combinatorial Vector Fields
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Added by
Marzieh Eidi
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We construct a Floer type boundary operator for generalised Morse-Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the $mathbb{Z}_2$ homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.
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Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invariant set. Likewise, considering a Morse decomposition omits much information about the individual Morse sets. In this paper, we propose a method to summarize changes in combinatorial dynamical systems by capturing changes in the so-called Conley-Morse graphs. A Conley-Morse graph contains information about both the structure of a selected Morse decomposition and about the Conley index at each Morse set in the decomposition. Hence, our method summarizes the changing structure of a sequence of dynamical systems at a finer granularity than previous approaches.
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We provide an intergral lift of the combinatorial definition of Heegaard Floer homology for nice diagrams, and show that the proof of independence using convenient diagrams adapts to this setting.
Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.
In [Janson & Marsden 2017] a dynamical system with a plastic self-organising velocity vector field was introduced, which was inspired by the architectural plasticity of the brain and proposed as a possible conceptual model of a cognitive system. Here we provide a more rigorous mathematical formulation of this problem, make several simplifying assumptions about the form of the model and of the applied stimulus, and perform its mathematical analysis. Namely, we explore the existence, uniqueness, continuity and smoothness of both the plastic velocity vector field controlling the observable behaviour of the system, and of the behaviour itself. We also analyse the existence of pullback attractors and of forward limit sets in such a non-autonomous system of a special form. Our results verify the consistency of the problem, which was only assumed in the previous work, and pave the way to constructing models with more sophisticated cognitive functions.
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