Templex: 一种连接同调与模板用于混沌吸引子的桥梁
Templex: a bridge between homologies and templates for chaotic attractors
AI总结：
Templex 通过结合细胞复形与有向图，为混沌吸引子提供更精确的拓扑描述和分类方法。
摘要：
同调理论引入细胞复形以提供空间的代数描述，直至拓扑等价。通过同调的分支 manifold 分析，可以在状态空间中研究吸引子：该策略从状态空间中的点云构建细胞复形，并利用同调群来描述其拓扑结构。然而，这种方法并未考虑流对细胞复形的作用。此处的程序被扩展以考虑这一基本属性，如通过模板实现。目标是通过在细胞复形上赋予一个有向图，以规定其最高维细胞之间的流方向。细胞复形与有向图的组合，称为 templex，被证明能够对混沌吸引子进行复杂的描述，并对其准确分类。对几个著名的混沌吸引子进行了研究——即螺旋和 funnel 罗斯勒吸引子、洛伦兹吸引子以及伯克和肖吸引子。与它们以模板形式的描述建立了联系。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
