# Dynamical Systems on Generalised Klein Bottles

## TL;DR

This paper introduces a new high-dimensional version of the Klein bottle and explores how it can be used to model complex dynamical systems in the human cortex.

## Contribution

The paper presents a novel generalisation of the Klein bottle and applies it to model cortical information processing with topological data analysis.

## Key findings

- High-dimensional Klein bottle generalisations can support continuous scalar fields and dynamical systems.
- Topological data analysis reveals potential symmetries in cortical information processing models.
- The approach offers new examples for studying complex dynamical systems on non-traditional state spaces.

## Abstract

We propose a high-dimensional generalisation of the standard Klein bottle extending 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.