CONICET-Universidad de Buenos Aires, Centro de Investigaciones del Mar y la Atmósfera (CIMA), C1428EGA CABA, Argentina.
Rouen Normandie University-CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France.
Chaos. 2022 Aug;32(8):083108. doi: 10.1063/5.0092933.
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.
同调理论引入胞腔复形，以提供拓扑等价意义下的空间的代数描述。通过同调的分岔流形分析可以研究状态空间中的吸引子：该策略从状态空间中的点云中构建胞腔复形，并使用同调群来刻画其拓扑结构。然而，该方法并未考虑流对胞腔复形的作用。在此，我们将该基本性质扩展到该方法中，就像在模板中所做的那样。目的是通过赋予胞腔复形一个有向图来实现，该有向图规定了其最高维胞腔之间的流动方向。将胞腔复形和有向图组合在一起，称为 templex，它可以对混沌吸引子进行精细的刻画，并对它们进行准确的分类。研究了几个著名的混沌吸引子的情况，即螺旋和漏斗型 Rössler 吸引子、Lorenz 吸引子、Burke 和 Shaw 吸引子以及一个四维系统。并与它们在模板中的描述建立了联系。
