Nilpotent centers of cubic systems

A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

“…However, this procedure is difficult to implement and with a high cost of computations to eliminate all the terms that not appear in the final expression of the normal form. Is for this reason that there are few families of nilpotent vector fields where the center conditions are known, see, for instance, [7,11,12,20,27]. Moreover, alternative methods are used to find such centers, as the generalized polar coordinates, Cherkas' method, approximation by nondegenerate systems, see, for instance, [6,16,17,[28][29][30]34,35].…”

Algebraic integrability of nilpotent planar vector fields

“…For example, Sadovskii [31] , Berthier and Moussu [11] , Andreev et al [10] , Giacomini et al [21] , García et al [20] , Algaba et al [4] have characterized the nilpotent centers. The center problem has been solved for some families, see [9,16,24] . Particularly, Chavarriga et al [13] and García and Giné [19] study the analytically integrable nilpotent centers.…”

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

“…According to Jules H. Poincare, a normal autonomous second-order differential system with polynomial right parts, in principle, allows its full qualitative investigation on an extended arithmetical plane R 2 x, y [1]. Inspired by the great Poincare's works, mathematicians of the next generations, including contemporary researchers, have studied some of such systems, for example, quadratic dynamic systems [2], ones containing nonzero linear terms, homogeneous cubic systems, and dynamic systems with nonlinear homogeneous terms of the odd degrees (3, 5, 7) [3], which have a center or a focus in a singular point O (0, 0) [4], as well as other particular kinds of systems.…”