ConleyDynamics.jl: A Julia package for multivector dynamics on Lefschetz complexes

81. Thomas Wanner:
    ConleyDynamics.jl: A Julia package for multivector dynamics on Lefschetz complexes
    Preprint, submitted for publication, 5 pages, 2024.

Abstract

Combinatorial topological dynamics is concerned with the qualitative study of dynamical behavior on discrete combinatorial structures. It was originally developed in the context of combinatorial vector fields (Forman, 1998a, 1998b), and has since been extended to combinatorial multivector fields (Lipinski et al., 2023; Mrozek, 2017) on arbitrary Lefschetz complexes (Lefschetz, 1942). For such systems, one can formulate a complete qualitative theory which includes notions of invariance, attractors, repellers, and connecting orbits. The global dynamical behavior is encoded in a Morse decomposition, and it can be studied further using algebraic topological tools such as the Conley index (Conley, 1978; Mischaikow & Mrozek, 2002) and connection matrices (Franzosa, 1989; Mrozek & Wanner, 2025). If the combinatorial multivector field is generated from a classical flow, one can derive statements about the underlying dyamics of the original system (Mrozek et al., 2022; Thorpe & Wanner, 2024a, 2024b). The Julia (Bezanson et al., 2017) package ConleyDynamics.jl provides computational tools for combinatorial topological dynamics, and should be of interest to both researchers and students which are curious about this emerging field.

Bibtex

@article{wanner:p24a,
   author = {Thomas Wanner},
   title = {Conley{D}ynamics.jl: {A} {J}ulia package for multivector
            dynamics on {L}efschetz complexes},
   journal = {Submitted for publication},
   year = 2024
   }