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

Abstract

The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by a large class of 3-dimensional nilpotent polynomial vector fields of arbitrary degree. In the discrete case, we prove that each dynamical system has a unique fixed point and there are 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, we prove that each dynamical system is polynomially integrable. Particularly, it is proved that the global dynamics of some low degree vector fields is completely understood and that there are invariant surfaces foliated by periodic orbits. As far as we know, this last property has not been shown before in the nilpotent context. We achieve our results by using the approach of polynomial automorphisms to obtain simplified conjugated dynamical systems, instead of considering only the usual linear transformations. Finally, we point out some similarities shared by the discrete and continuous dynamical systems, and we formulate some open questions motivated by our results, which are related with the Markus–Yamabe conjecture and the problem of planar limit cycles.