Unbounded dynamics for vector fields

TL;DR

This paper investigates the unbounded dynamics of three-dimensional vector fields with finitely many fixed points, providing conditions for unbounded invariant manifolds and applying these to specific dynamical systems.

Contribution

It establishes sufficient conditions for unbounded invariant manifolds in vector fields and applies these results to well-known dynamical systems.

Findings

Conditions for unbounded invariant manifolds are derived.

Applications to Genesio-Tesi, Belousov-Zhabotinsky, and Michelson systems.

Insights into the unbounded dynamics of these systems.

Abstract

Consider a three-dimensional vector field $F$ which generates a finite number of fixed points - what can we say on its unbounded dynamics? In this paper we tackle this question, and prove sufficient conditions for $F$ to have fixed points with unbounded invariant manifolds. Following that, we use these results to study the dynamics of the Genesio-Tesi system, the Belousov-Zhabotinsky reaction, and the Michelson system.