Injectivity and Almost Global Stability of Hurwitz Vector Fields

TL;DR

This paper surveys key conjectures in vector field theory and introduces new examples of vector fields with almost global stability using the innovative concept of density functions.

Contribution

It provides a survey of the Jacobian and Markus–Yamabe conjectures and introduces new stable vector field examples via density functions.

Findings

Examples related to the Jacobian conjecture and Markus–Yamabe conjecture.

Construction of vector fields with almost global stability.

Application of density functions to stability analysis.

Abstract

We present, in dimension $n \geq 3$, a survey of examples to: the Jacobian conjecture, the weak Markus--Yamabe conjecture. Furthermore, we show and construct new examples of vector fields where the origin is almost globally asymptotically stable by using the novel concept of density functions introduced by Rantzer.