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Finding attracting sets using combinatorial multivector fields

Justin Thorpe, Thomas Wanner

TL;DR

The paper develops methods to identify attracting sets in continuous dynamics by using combinatorial multivector fields (CMVFs) on Lefschetz complexes as a finite combinatorial representation of phase space dynamics. A CMVF partitions a Lefschetz complex $X$ into locally closed sets and encodes transitions via a multivalued flow $\\Pi_{\\mathcal{V}}$, absorbing transversality into the CMVF. It then builds Morse decompositions and a connection-matrix framework to extract Conley indices and attractors, with demonstrations on planar and 3D Allen-Cahn projections and a Julia-based implementation. The approach yields a computable, algebraic route to certify attracting regions and connecting orbits in complex dynamics, and integrates with Lyapunov-function approximations for rigorous existence proofs.

Abstract

We discuss the identification of attracting sets using combinatorial multivector fields (CMVF) from Conley-Morse-Forman theory. A CMVF is a dynamical system induced by the action of a continuous dynamical system on a phase space discretization that can be represented as a Lefschetz complex. There is a rich theory under development establishing the connections between the induced and underlying dynamics and emphasizing computability. We introduce the main ideas behind this theory and demonstrate how it can be used to identify regions of interest within the global dynamics via graph-based algorithms and the connection matrix.

Finding attracting sets using combinatorial multivector fields

TL;DR

The paper develops methods to identify attracting sets in continuous dynamics by using combinatorial multivector fields (CMVFs) on Lefschetz complexes as a finite combinatorial representation of phase space dynamics. A CMVF partitions a Lefschetz complex into locally closed sets and encodes transitions via a multivalued flow , absorbing transversality into the CMVF. It then builds Morse decompositions and a connection-matrix framework to extract Conley indices and attractors, with demonstrations on planar and 3D Allen-Cahn projections and a Julia-based implementation. The approach yields a computable, algebraic route to certify attracting regions and connecting orbits in complex dynamics, and integrates with Lyapunov-function approximations for rigorous existence proofs.

Abstract

We discuss the identification of attracting sets using combinatorial multivector fields (CMVF) from Conley-Morse-Forman theory. A CMVF is a dynamical system induced by the action of a continuous dynamical system on a phase space discretization that can be represented as a Lefschetz complex. There is a rich theory under development establishing the connections between the induced and underlying dynamics and emphasizing computability. We introduce the main ideas behind this theory and demonstrate how it can be used to identify regions of interest within the global dynamics via graph-based algorithms and the connection matrix.

Paper Structure

This paper contains 10 sections, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Lefschetz Complexes
  3. Lefschetz Homology
  4. Multivector Fields via Phase Space Discretizations
  5. Combinatorial Dynamics and the Conley Index
  6. Morse Decompositions
  7. Connection Matrices
  8. Examples
  9. A Planar Gradient System
  10. The Three-Dimensional Allen-Cahn Projection

Figures (5)

  • Figure 1: Morse sets for the planar sample system, based on a random Delaunay triangulation.
  • Figure 2: Morse interval which includes all stable and index one stationary states of the planar sample system.
  • Figure 3: Four sample Morse intervals for the planar system with nine equilibria.
  • Figure 4: Enclosures for the seven equilibrium solutions of the three-dimensional sample system.
  • Figure 5: A Morse interval for the three-dimensional sample system, which encloses a index $1$ equilibrium, two stable stationary states, and the two heteroclinics between these states.