从数据中混沌拓扑的函子不变量
Functorial invariants for chaos topology from data
AI总结：
本文通过引入有向路径代数和边缘算子，提出混沌动力学的函子不变量，用于识别临界点和验证数据驱动模型。
摘要：
templex是一种连接同调与混沌动力学模板的拓扑对象。本文将templex置于范畴论中，引入有向路径代数、有向路径上的边缘算子以及区别于有向同调的有向循环等价关系。所得的函子不变量分为两种：阿贝尔群不变量，即同调群，和半群不变量，即生成半群。这些不变量通过遗忘函子分离，并构成一个稳健的框架，用于识别临界点、区分物理机制以及将数据驱动模型与观测或模拟进行基准测试。该方法提出了一种非度量标准来从有限时间数据中定义混沌，并揭示了拓扑变化模式的可连接性是由于有向路径代数的半群结构直接导致的。罗素和洛伦兹吸引子作为范例被提出，随后分析了气候模拟和实验语音信号。
英文摘要：
The templex is a topological object bridging homologies and templates for chaotic dynamics. This article places the templex within category theory, introducing a directed path algebra, an edge operator on directed paths, and an equivalence relation for directed cycles that is distinct from directed homologies. The resulting functorial invariants are of two kinds: abelian-group invariants, namely the homology groups, and semigroup invariants, namely the generatex semigroups. These invariants are separable through forgetful functors and constitute a robust framework for identifying tipping points, disambiguating physical mechanisms, and benchmarking data-driven models against observations or simulations. The formulation sets forth a non-metric criterion for chaos from finite-time data and reveals that the concatenable nature of Topological Modes of Variability is a direct consequence of the semigroup structure of the directed path algebra. Two applications are presented: an experimental speech signal and a climatic numerical simulation.
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。
