Connecting Curves in Higher Dimensions

TL;DR

This paper extends the concept of connecting curves, which organize the structure of strange attractors, to higher-dimensional dynamical systems, providing theoretical properties, selection rules, and illustrative examples.

Contribution

It introduces a generalized framework for connecting curves in higher dimensions, including derivation of properties and selection rules, with practical examples for dimensions 3 to 5.

Findings

Derived general properties of connecting curves in higher dimensions

Established selection rules for connecting curves

Provided illustrative examples for systems of dimension 3, 4, and 5

Abstract

Connecting curves have been shown to organize the rotational structure of strange attractors in three-dimensional dynamical systems. We extend the description of connecting curves and their properties to higher dimensions within the special class of differential dynamical systems. The general properties of connecting curves are derived and selection rules stated. Examples are presented to illustrate these properties for dynamical systems of dimension n=3,4,5.