Conley-Morse-Forman theory for combinatorial multivector fields
This work provides a theoretical foundation for algorithmic analysis of dynamical systems via combinatorialization, benefiting researchers in dynamical systems and computational topology.
The paper introduces combinatorial multivector fields, generalizing Forman's combinatorial vector fields, and defines key dynamical concepts such as isolated invariant sets, Conley index, and Morse decompositions. It provides a topological characterization of attractors and repellers and proves Morse inequalities, with a prototype algorithm for algorithmic analysis of dynamical systems.
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.