Combinatorial vs. classical dynamics: Recurrence

TL;DR

This paper introduces a combinatorial approach to establish the existence of periodic orbits in classical dynamical systems, enabling computer-assisted proofs and analysis of chaotic behavior.

Contribution

It presents a general existence theorem for periodic orbits in semiflows using combinatorial topological methods, bridging classical and combinatorial dynamics.

Findings

Provides a framework for computer-assisted proofs of periodic orbits.

Demonstrates how combinatorial techniques can verify assumptions in dynamical systems.

Enables analysis of chaotic behavior through combinatorial methods.

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.