Discrete and continuous dynamics of real $3$-dimensional nilpotent polynomial vector fields

TL;DR

This paper studies the dynamics of real 3D nilpotent polynomial vector fields, revealing fixed point properties, integrability, and global behavior, with implications for longstanding conjectures and cycle problems.

Contribution

It provides new results on fixed points, integrability, and global dynamics of nilpotent polynomial vector fields in three dimensions, including conditions for integrability and existence of invariant surfaces.

Findings

Unique fixed point in discrete systems

No 2-cycles in discrete systems

Polynomial integrability in continuous systems

Abstract

A large class of real $3$-dimensional nilpotent polynomial vector fields of arbitrary degree is considered. The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by these vector fields. In the discrete case, it is proved that each dynamical system has a unique fixed point and no $2$-cycles. Moreover, either the fixed point is a global attractor or there exists a $3$-cycle which is not a repeller. In the continuous setting, it is proved that each dynamical system is polynomially integrable. In addition, for a subclass of the considered vector fields, the system is polynomially completely integrable. Furthermore, for a family of low degree vector fields, it is provided a more precise description about the global dynamics of the trajectories of the induced dynamical system. In particular, it is proved the existence of an invariant surface foliated by periodic orbits. Finally, some remarks and open questions, motivated by our results, the Markus--Yamabe Conjecture and the problem of planar limit cycles, are given.