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Coarse chain recurrence, Morse graphs with finite errors, and persistence of circulations

Tomoo Yokoyama

TL;DR

The paper develops a unified framework that extends chain recurrence and Morse graphs to coarse, finite-energy perturbations, producing one-parameter filtrations CR_ε and CR^Σ_ε and their ℓ^p variants. It introduces ε-ℓ^p-chain recurrence and ε-ℓ^p chain recurrence diagrams, along with Morse graphs and hyper-graphs with ε- and (ε,ν)-errors, plus vertex/graph collapses that encode how recurrent structures persist or dissolve as perturbation scales vary. The authors establish singular-limit behaviors and connections to difference equations, and illustrate the theory with extensive examples, including control-based flows (finite-energy perturbations) and persistence of recurrence. The framework provides new tools for analyzing reachability, robustness, and the effect of bounded perturbations on dynamical systems, with potential applications to numerical analysis, geodesic flows, and beyond.

Abstract

This paper provides a unified framework connecting dynamical systems with tools from topological data analysis and geometric topology and inspires new interactions among dynamical systems, topology, and nonlinear analysis. To this end, we introduce a one-parameter family of ``chain recurrences'' that generalizes chain recurrence and induces a natural filtration on the underlying metric space of a dynamical system. In particular, the forward directions of the filtrations characterize the level of control required to return to the original position, and the backward directions capture the robustness of the recurrence. The resulting filtrations yield potentials and bifurcation diagrams of dynamical systems that encode the evolution of recurrent sets under bounded total or stepwise perturbations. In addition, we extend Morse graphs to one-parameter families of ``coarse Morse graphs,'' which evolve through vertex collapses reflecting coarse recurrence transitions. These constructions not only refine Conley's decomposition but also reveal singular limit behaviors as the perturbation level vanishes. Furthermore, we establish analogous filtrations for difference equations to bridge the theoretical framework with numerical analysis.

Coarse chain recurrence, Morse graphs with finite errors, and persistence of circulations

TL;DR

The paper develops a unified framework that extends chain recurrence and Morse graphs to coarse, finite-energy perturbations, producing one-parameter filtrations CR_ε and CR^Σ_ε and their ℓ^p variants. It introduces ε-ℓ^p-chain recurrence and ε-ℓ^p chain recurrence diagrams, along with Morse graphs and hyper-graphs with ε- and (ε,ν)-errors, plus vertex/graph collapses that encode how recurrent structures persist or dissolve as perturbation scales vary. The authors establish singular-limit behaviors and connections to difference equations, and illustrate the theory with extensive examples, including control-based flows (finite-energy perturbations) and persistence of recurrence. The framework provides new tools for analyzing reachability, robustness, and the effect of bounded perturbations on dynamical systems, with potential applications to numerical analysis, geodesic flows, and beyond.

Abstract

This paper provides a unified framework connecting dynamical systems with tools from topological data analysis and geometric topology and inspires new interactions among dynamical systems, topology, and nonlinear analysis. To this end, we introduce a one-parameter family of ``chain recurrences'' that generalizes chain recurrence and induces a natural filtration on the underlying metric space of a dynamical system. In particular, the forward directions of the filtrations characterize the level of control required to return to the original position, and the backward directions capture the robustness of the recurrence. The resulting filtrations yield potentials and bifurcation diagrams of dynamical systems that encode the evolution of recurrent sets under bounded total or stepwise perturbations. In addition, we extend Morse graphs to one-parameter families of ``coarse Morse graphs,'' which evolve through vertex collapses reflecting coarse recurrence transitions. These constructions not only refine Conley's decomposition but also reveal singular limit behaviors as the perturbation level vanishes. Furthermore, we establish analogous filtrations for difference equations to bridge the theoretical framework with numerical analysis.

Paper Structure

This paper contains 67 sections, 47 theorems, 90 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statements of main results
  3. Preliminaries
  4. Basic concepts in set theory
  5. Basic concepts in combinatorics
  6. Basic concepts in mappings
  7. Variant of chain recurrence
  8. Basic concepts for homeomorphisms
  9. Morse graph for homeomorphisms
  10. Basic concepts for semiflows
  11. Chain recurrence and relative concepts for semiflows
  12. Variant of the chain recurrence
  13. Basic concepts for flows
  14. Coarse chain recurrence for mappings
  15. $\varepsilon$-$\ell^p$-chains and $\ell^p$-chain recurrence
  16. ...and 52 more sections

Key Result

Theorem 1

For any continuous mapping $f \colon X \to X$ on a metric space $X$, the family $(\mathrm{CR}_{\varepsilon}(f))_{\varepsilon \in \mathbb{R} }$ (see Definitions def:reachable, def:cr, def:cr2, def:cr02, and def:cr2_neg below) is a filtration with $\mathrm{CR}(f) = \mathrm{CR}_{0}(f)$ and $X = \bigcup

Figures (6)

  • Figure 1: Diagrams $D_{\mathrm{CR}}(f_{R})$ and $D_{\mathrm{CR}}^\Sigma(f_{R})$.
  • Figure 2: Diagrams $D_{\mathrm{CR}}(f_{\mathrm{rep}})$, $D_{\mathrm{CR}}^\Sigma(f_{\mathrm{rep}})$, and $D^{\ell^p}_{\mathrm{CR}}(f_{\mathrm{att}})$.
  • Figure 3: A schematic picture of the $\sum$chain recurrent diagram of a mapping $f$ defined in § \ref{['sec:c_ex_001']}.
  • Figure 4: Diagrams $D_{\mathrm{CR}}(v_{\mathrm{rep}})$, $D_{\mathrm{CR}}^\Sigma(v_{\mathrm{rep}})$, and $D_{\mathrm{CR}}(v_{\mathrm{att}}) = D_{\mathrm{CR}}^\Sigma(v_{\mathrm{att}})$.
  • Figure 5: A schematic picture of the $\sum$chain recurrent diagram of a flow $v$.
  • ...and 1 more figures

Theorems & Definitions (129)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 119 more