Abstract

Topological invariants in dynamics such as the fixed point index or the Conley index have many applications in the qualitative study of differential equations and discrete dynamical systems. In particular they, are used in the existence proofs of stationary and periodic orbits, homoclinic connections and chaotic invariant sets. The classical methods require analytic formulas for vector fields or maps generating the dynamics. This is an obstacle in the case of dynamics known only from finite samples gathered from observations, experiments or evaluations. In 1998, in his seminal work on discrete Morse theory, R. Forman introduced and studied purely combinatorial analogues of a vector fields and the associated flows. It turns out that many concepts of classical topological dynamics may be adapted to the combinatorial setting by following and generalizing the ideas of Forman. The resulting combinatorial topological dynamics provides new tools in the study of sampled dynamics. It also helps automatize rigorous numerics of dynamical systems.  In the talk I will first recall some classical results on which combinatorial dynamics is based and then I will overview some recent results in the emerging new field. 

Outline

Classics of topological dynamics: Ważewski Theorem, Conley index
Challenges of sampled dynamical systems
Combinatorial version of Morse theory by Forman
Conley theory for combinatorial multivector fields
Some applications to rigorous numerics in dynamics