The Discrete and Continuous Markus-Yamabe Stability Conjectures

TL;DR

This paper investigates the Markus-Yamabe Conjecture for polynomial vector fields in R^n, exploring conditions under which the origin is a global attractor or where orbits escape to infinity, with new examples and counterexamples.

Contribution

It provides new classes of vector fields that either confirm or challenge the conjecture, especially in the case of linearly dependent or independent Jacobian rows.

Findings

Existence of non-linearly triangularizable vector fields with global attractor at origin

Construction of vector fields with orbits escaping to infinity

Identification of vector fields with period-3 points in the discrete case

Abstract

We study the discrete and continuous versions of the Markus- Yamabe Conjecture for polynomial vector fields in R^n (especially when n = 3) of the form X = \lambda I+H where \lambda is a real number, I the identity map, and H a map with nilpotent Jacobian matrix JH. We consider the case where the rows of JH are linearly dependent over R and that where they are linearly independent over R. In the former, we find non-linearly triangularizable vector fields X for which the origin is a global attractor for both the continuous and the discrete dynamical systems generated by X. In the independent continuous case, we present a family of vector fields which have orbits escaping to infinity. In the independent discrete case, we present a large family of vector fields which have a periodic point of period 3.