On the non-integrability and dynamics of discrete models of threads

TL;DR

This paper investigates the dynamics of discrete thread models, showing they are generally non-integrable and analyzing conditions for complex behavior like positive topological entropy.

Contribution

It demonstrates non-integrability of discrete thread systems under weak assumptions and provides conditions for positive topological entropy.

Findings

Discrete models of threads are non-integrable in the Liouville sense.

Conditions for positive topological entropy are established.

Analysis includes both free and externally forced thread models.

Abstract

In the paper, we study the dynamics of planar $n$-gons, which can be considered as discrete models of threads. The main result of the paper is that, under some weak assumptions, these systems are not integrable in the sense of Liouville. This holds for both completely free threads and for threads with fixed points that are placed in external force fields. We present sufficient conditions for the positivity of topological entropy in such systems. We briefly consider other dynamical properties of discrete threads and we also consider discrete models of inextensible yet compressible threads.