Dynamical Systems On Generalised Klein Bottles

TL;DR

This paper introduces a high-dimensional generalization of the Klein bottle and explores dynamical systems on these spaces, focusing on their potential to model cortical information processing and exhibit Klein bottle symmetries.

Contribution

It presents a novel high-dimensional Klein bottle generalization and develops methods to generate scalar fields and dynamical systems on these spaces.

Findings

Topological data analysis reveals complex dynamical behaviors.

Potential Klein bottle symmetries observed in cortical models.

Framework for future exploration of high-dimensional topological dynamics.

Abstract

We propose a high dimensional generalisation of the standard Klein bottle, going beyond those considered previously. We address the problem of generating continuous scalar fields (distributions) and dynamical systems (flows) on such state spaces, which can provide a rich source of examples for future investigations. We consider a class of high dimensional dynamical systems that model distributed information processing within the human cortex, which may be capable of exhibiting some Klein bottle symmetries. We deploy topological data analytic methods in order to analyse their resulting dynamical behaviour, and suggesting future challenges.