The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

[摘要] In this article we determine the global geometry of the planarquadratic differential systems with a weak focus of third order. Thisclass plays a significant role in the context of Hilbert's 16-thproblem. Indeed, all examples of quadratic differential systems withat least four limit cycles, were obtained by perturbing a system inthis family. We use the algebro-geometric concepts of divisor andzero-cycle to encode global properties of the systems and to givestructure to this class. We give a theorem of topologicalclassification of such systems in terms of integer-valued affineinvariants. According to the possible values taken by them in thisfamily we obtain a total of $18$ topologically distinct phaseportraits. We show that inside the class of all quadratic systemswith the topology of the coefficients, there exists a neighborhood ofthe family of quadratic systems with a weak focus of third order andwhich may have graphics but no polycycle in the sense of cite{DRR}and no limit cycle, such that any quadratic system in thisneighborhood has at most four limit cycles.

[学科分类] 数学（综合）