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Templex: a bridge between homologies and templates for chaotic attractors

Gisela D. Charó, Christophe Letellier, Denisse Sciamarella

TL;DR

The paper addresses the gap between homology-based descriptions of chaotic attractors and template-based classifications by introducing templex, a structure that couples a BraMAH complex with a flow-informed digraph. The approach provides a practical workflow: build a BraMAH complex from state-space data, orient it according to the flow, and augment it with a digraph to form a templex $T=(K,G)$; extract generating templexes, stripexes, and generatexes to recover template-like decompositions and invariants. The authors demonstrate the method on the Lorenz, Rössler, three-strip Rössler, and Burke–Shaw attractors, obtaining generating templexes that reproduce the corresponding templates and reveal detailed flow structure (e.g., joining loci, twists, and strip organization) while enriching the description with homological data. This bridging of homology and templates provides a robust framework for higher-dimensional chaotic attractors and suggests a path toward topological chaos analysis beyond the classical three-dimensional setting.

Abstract

The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies: this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated -- namely the spiral and funnel Rössler attractors, the Lorenz attractor and the Burke and Shaw attractor. A link is established with their description in terms of templates.

Templex: a bridge between homologies and templates for chaotic attractors

TL;DR

The paper addresses the gap between homology-based descriptions of chaotic attractors and template-based classifications by introducing templex, a structure that couples a BraMAH complex with a flow-informed digraph. The approach provides a practical workflow: build a BraMAH complex from state-space data, orient it according to the flow, and augment it with a digraph to form a templex ; extract generating templexes, stripexes, and generatexes to recover template-like decompositions and invariants. The authors demonstrate the method on the Lorenz, Rössler, three-strip Rössler, and Burke–Shaw attractors, obtaining generating templexes that reproduce the corresponding templates and reveal detailed flow structure (e.g., joining loci, twists, and strip organization) while enriching the description with homological data. This bridging of homology and templates provides a robust framework for higher-dimensional chaotic attractors and suggests a path toward topological chaos analysis beyond the classical three-dimensional setting.

Abstract

The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies: this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated -- namely the spiral and funnel Rössler attractors, the Lorenz attractor and the Burke and Shaw attractor. A link is established with their description in terms of templates.
Paper Structure (13 sections, 1 theorem, 39 equations, 20 figures, 1 table)

This paper contains 13 sections, 1 theorem, 39 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. Topological background
  3. Templates as knot-holders for chaotic attractors
  4. Homology groups
  5. Templex
  6. Characterization of 3D chaotic attractors
  7. The Lorenz attractor
  8. The Rössler attractor
  9. The three-strips Rossler attractor
  10. The Burke and Shaw attractor
  11. Conclusion
  12. Computing the homology groups of the torus
  13. From the direct template to the reduced one

Key Result

Theorem 1

Let $C_k^{\alpha_1}$ and $C_k^{\alpha_2}$ be two $k$-chains $\in$${\cal C}_k$. They are said to be homologous --- noted $C_k^{\alpha_1} \sim C_k^{\alpha_2}$ --- if and only if there exists a boundary of a ($k+1$)-chain $C_{k+1}$ such that $\partial_{k+1}(C_{k+1})=C_k^{\alpha_1} - C_k^{\alpha_2}$.

Figures (20)

  • Figure 1: Bounding tori of various genus $g$ ($g \leqslant 4$). The $g-1$ components of the Poincaré section are plotted as thick lines. Case (a) applies for the Rössler attractor and (b) for the Lorenz attractor. Cases (c) may correspond to the 3-fold covers of the proto-Lorenz system.Mir93Let95d
  • Figure 2: An ingoing strip is split into three outgoing strips according to the two critical points P$_1$ and P$_2$ (a) which are then joined into a single outgoing strip (b). In (a), true boundaries are in solid thick lines and fictive boundaries are in dashed thick lines. The very thick line (b) represents the joining line (branch line).
  • Figure 3: An example of template made of $N_{\rm s} = 3$ strips and $N_{\rm loc} = 1$ joining line. This template corresponds to an attractor bounded by a genus-1 torus. True boundaries are in solid line and fictive boundaries are in dashed line. The thick line represents the joining line. The arrow indicates the direction of the flow $\phi_t$.
  • Figure 4: (a-b) Templex $T_1({\rm L})=(K_1({\rm L}),G_1({\rm L}))$ and (c-d) generating templex $T_2({\rm L})=(K_2({\rm L}),G_2({\rm L}))$ for the Lorenz attractor. In (e), digraph $G_2'({\rm L})$ corresponding to another generating complex $K_2'({\rm L})$ --- not shown --- for the Lorenz attractor. Ingoing and outgoing nodes are squared and circled, respectively.
  • Figure 5: Orientation of the normal to a 2-cell to define relative top (and bottom). This is required for determining the order with which 2-cells are joined at a joining locus (see below).
  • ...and 15 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5