论文标题
真正的3 $二维nilpotent多项式矢量字段的离散和连续动态
Discrete and continuous dynamics of real $3$-dimensional nilpotent polynomial vector fields
论文作者
论文摘要
考虑了一大批真正的$ 3 $维度nilpotent多项式矢量字段。这项工作的目的是介绍这些向量场引起的离散和连续动力学系统的一般特性。在离散的情况下，证明每个动态系统都有一个唯一的固定点，没有$ 2 $ CYCLE。此外，固定点是全球吸引子，或者存在一个$ 3 $ cycle，这不是驱虫者。在连续的环境中，证明每个动态系统都是多项式集成的。另外，对于考虑到所考虑的矢量场的子类，系统在多项式上是完全可集成的。此外，对于一个低度矢量场的家庭，它得到了有关诱导动力系统轨迹的全局动力学的更精确描述。特别是，它被证明是由周期轨道散落的不变表面的存在。最后，给出了一些言论和开放性问题，在我们的结果中引起的，马克斯 - 雅amabe的猜想和平面极限周期的问题。
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.
