Canadian Mathematical Society
Canadian Mathematical Society
  location:  Publicationsjournals
Search results

Search: All MSC categories starting with 51F14

  Expand all        Collapse all Results 1 - 1 of 1

1. CJM 2004 (vol 56 pp. 310)

Llibre, Jaume; Schlomiuk, Dana
The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order
In this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of $18$ topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of \cite{DRR} and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

Categories:34C40, 51F14, 14D05, 14D25

© Canadian Mathematical Society, 2014 :