Dynamical and algebraic analysis of planar polynomial vector fields linked to orthogonal polynomials

Abstract

In the present work, our goal is to establish a study of some families of quadratic polynomial vector fields connected to orthogonal polynomials that relate, via two different points of view, the qualitative and the algebraic ones. We extend those results that contain some details related to differential Galois theory as well as the inclusion of Darboux theory of integrability and the qualitative theory of dynamical systems. We conclude this study with the construction of differential Galois groups, the calculation of Darboux first integral, and the construction of the global phase portraits

在当前的工作中，我们的目标是建立与正交多项式相关的二次多项式矢量场的一些族的研究 ，该正交多项式通过两种不同的观点相互关联，即定性和代数。 我们扩展了这些结果，这些结果 包含与微分加洛瓦理论有关的一些细节，以及包含达布可积性理论和动力学系统的定性理论。我 们以差分伽罗瓦群的构造，达布克斯第一积分的计算以及整体相像的构造来结束本研究。