ConleyDynamics.jl

Conley index and multivector fields for Julia.

Introduction

ConleyDynamics.jl is a Julia package for studying combinatorial multivector fields using Conley theory. The multivector fields can be studied on arbitrary Lefschetz complexes, which include both simplicial and cubical complexes as important special cases. The concept of combinatorial multivector field generalizes Forman vector fields, which were originally introduced to study Morse theory in a discrete combinatorial setting.

Features

• Data structures for Lefschetz complexes, in particular simplicial and cubical complexes.
• Classical Forman combinatorial vector fields and multivector fields are supported.
• Computation of Conley indices, connection matrices, and Conley-Morse graphs.
• Basic homology algorithms over finite fields and the rationals, including persistent homology and relative homology.
• Algorithms rely on a built-in sparse matrix implementation which is geared towards computations over finite fields and the rationals.

Installation

To use ConleyDynamics.jl please install Julia 1.10 or higher. See https://julialang.org/downloads/ for instructions on how to obtain Julia for your system.

At the Julia prompt simply type

julia> using Pkg; Pkg.add("ConleyDynamics")

After Julia has finished downloading and precompiling the package and all of its dependencies, you can start using it by typing

julia> using ConleyDynamics

Whenever possible, the functions implemented in ConleyDynamics.jl have been parallelized, so as to speed up performance on multi-core computers. This, however, is only taken advantage of if Julia is started with multiple threads. For example, if you would like to start Julia using 8 threads, use the command

julia -t 8

or

julia --threads=8

If you want to leave the number of threads selection to the machine, simply use

julia --threads=auto

Once you are in the Julia REPL, you can see the number of used threads using the command

julia> Threads.nthreads()

Manual Outline

The Tutorial briefly explains how to get started with ConleyDynamics.jl. More details, including on the underlying mathematics, are provided in the following three sections, which cover Lefschetz complexes, homology, and Conley theory including connection matrices. After a discussion of all included examples in the Examples section, the manual concludes with a description of the sparse matrix format underlying the package.

• Tutorial
• • Creating Simplicial Complexes
  • Computing Homology and Persistence
  • Forman Vector Fields
  • Isolated Invariant Sets
  • Connection Matrices
  • Multivector Fields
  • Analyzing Planar Vector Fields
  • References
• Lefschetz Complexes
• • Basic Lefschetz Terminology
  • Lefschetz Complex Data Structure
  • Simplicial Complexes
  • Cubical Complexes
  • Lefschetz Complex Operations
  • References
• Homology
• • Lefschetz Complex Homology
  • Relative Homology
  • Persistent Homology
  • References
• Conley Theory
• • Multivector Fields
  • Invariance and Conley Index
  • Morse Decompositions
  • Connection Matrices
  • Extracting Subsystems
  • Analysis of a Planar System
  • Analysis of a Spatial System
  • Forman's Morse Complex
  • References
• Examples
• • A One-Dimensional Forman Field
  • A Planar Forman Vector Field
  • The Multivector Field from the Logo
  • Critical Flow on a Simplex
  • Flow on a Cylinder and a Moebius Strip
  • Nonunique Connection Matrices
  • Forcing Three Connection Matrices
  • A Lefschetz Multiflow Example
  • Small Complex with Periodicity
  • Subdividing a Multivector
  • A Combinatorial Lorenz System
  • Chaos in the Dunce Hat
  • Chaos in a Space with Torsion
  • References
• Sparse Matrices
• • Sparse Matrix Format
  • Creating Sparse Matrices
  • Sparse Matrix Access
  • Elementary Matrix Operations
  • Sparse Matrix Information

Citation

If you use ConleyDynamics.jl, please cite it. There is a JOSS paper published at https://doi.org/10.21105/joss.08085, see [Wan25]. The BibTeX entry for this paper is:

@article{wanner:25a,
   author = {Thomas Wanner},
   title = {Conley{D}ynamics.jl: {A} {J}ulia package for multivector
            dynamics on {L}efschetz complexes},
   journal = {Journal of Open Source Software},
   doi = {10.21105/joss.08085},
   volume = {10},
   number = {111},
   pages = {8085},
   url = {https://joss.theoj.org/papers/10.21105/joss.08085},
   year = {2025}
   }

License

ConleyDynamics.jl is provided under an MIT license.