Connecting curves for dynamical systems

TL;DR

This paper introduces connecting curves as one-dimensional sets that describe and constrain the integral curves of dynamical systems, providing richer information than fixed points alone.

Contribution

It presents a novel concept of connecting curves, with two equivalent definitions from dynamical systems and differential geometry, and demonstrates how to compute and analyze them.

Findings

Connecting curves pass through fixed points.

They offer additional insights into system dynamics.

Illustrations on various dynamical systems show their properties.

Abstract

We introduce one dimensional sets to help describe and constrain the integral curves of an $n$ dimensional dynamical system. These curves provide more information about the system than the zero-dimensional sets (fixed points) do. In fact, these curves pass through the fixed points. Connecting curves are introduced using two different but equivalent definitions, one from dynamical systems theory, the other from differential geometry. We describe how to compute these curves and illustrate their properties by showing the connecting curves for a number of dynamical systems.