Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes
Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes
复制标题
DOI：
10.1007/s10208-016-9330-z
复制
发表时间：
2015-05
影响因子：
3
通讯作者：
M. Mrozek
中科院分区：
文献类型：
--
作者：
M. Mrozek
引入组合多向量场，将其与多值动力学联系起来，研究其拓扑特征。我们的组合多向量场推广了Forman的组合向量场。我们定义了孤立不变集、Conley指数、吸引子、排斥子和Morse分解。我们给出了吸引子和排斥子的拓扑特征，并证明了莫尔斯不等式。该推广旨在通过微分方程给出的流动组合和物理和数值实验中的采样动力学来对动力系统进行算法分析。我们为这些应用提供了一个原型算法。
We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.