Combinatorial vs. classical dynamics: Recurrence

72. Marian Mrozek, Roman Srzednicki, Justin Thorpe, Thomas Wanner:
    Combinatorial vs. classical dynamics: Recurrence
    Communications in Nonlinear Science and Numerical Simulation 108, Paper No. 106226, 30 pages, 2022.

Abstract

Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in discrete contexts, such as graph theory or in the recently developed field of combinatorial dynamics, is straightforward and computationally feasible. In this paper, we present an approach to study classical dynamical systems as given by semiflows or flows using techniques from combinatorial topological dynamics. More precisely, we present a general existence theorem for periodic orbits of semiflows which is based on suitable phase space decompositions, and indicate how combinatorial techniques can be used to satisfy the necessary assumptions. In this way, one can obtain computer-assisted proofs for the existence of periodic orbits and even certain chaotic behavior.

Links

The preprint version of the paper can be downloaded from https://arxiv.org/abs/2108.13978, while the published version of the paper can be found at https://doi.org/10.1016/j.cnsns.2021.106226.

Bibtex

@article{mrozek:etal:22a,
   author = {Marian Mrozek and Roman Srzednicki and Justin Thorpe and Thomas Wanner},
   title = {Combinatorial vs. classical dynamics: {R}ecurrence},
   journal = {Communications in Nonlinear Science and Numerical Simulation},
   year = 2022,
   volume = {108},
   pages = {Paper {N}o. 106226, 30 pages},
   doi = {10.1016/j.cnsns.2021.106226}
   }