Templex: A bridge between homologies and templates for chaotic attractors

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, the Burke and Shaw attractor, and a four-dimensional system. A link is established with their description in terms of templates.

Review Article: Dynamical Systems, Algebraic Topology, and the Climate Sciences

“…In Sec. 3.3, we introduce the templex, a novel concept in algebraic topology (Charó et al, 2022a;Charó et al, 2022b), which complements the previously mentioned cell complexes of BraMAH by a directed graph (digraph), whose nodes are the cells and which approximates the flow on the branched manifold. The extension of this concept to the noise-perturbed chaotic attractors of RDS theory follows in Sec.…”

“…Including the arrow of time into the description calls for a more refined mathematical object, in which the topological properties of a flow in phase space come into light through the combined analysis of both the spatial structure of the underlying branched manifold and of the semi-flow upon it. Charó et al (2022a) introduced such a novel type of mathematical object and called it a templex, a word obtained from the contraction between "template" and "complex." A template in dynamical systems theory is a synonym for a knot-holder (Birman and Williams, 1983a;Tufillaro et al, 1992;Ghrist et al, 1997).…”

“…It is achieved through a set of well defined operations that include flow-orienting the cell complex; minimizing the cell structure at the joining loci, where the tearing of the flow takes place, to obtain a generating templex; calculating the cycles of the digraph; and checking for local twists, since uneven torsions in a strip correspond to a local twist in a stripex. The reader is referred to the steps in Charó et al (2022a) for further details. This dissection of the cell complex into stripexes provides the information that enables one to distinguish the topological properties of the two Rössler attractors from each other.…”

“…Combining the cell complex and the digraph, we can define and algebraically compute the stripexes. For details on this procedure, the reader is again referred to Charó et al (2022a). The stripexes for the spiral attractor are given by two paths along the cell complex, indicated by the two cycles below, the first of which is twisted.…”

“…1000 Acknowledgements. Sections 3.2-3.4 of this article rely to a great extent on recent joint work with Guillermo Artana, Gisela D. Charó, Mickaël Chekroun and Christophe Letellier (Charó et al, 2019(Charó et al, , 2021aCharó et al, 2021b;Charó et al, 2022a;Charó et al, 2022b).…”