The interplay among the topological bifurcation diagram, integrability and geometry for the family QSH(D)

Author(s)

Mota, Marcos C. ; Oliveira, Regilene ; Travaglini, Ana Maria

• Document type: Journal article
• Source: Geometriae Dedicata; v. 217, n. 6, p. 42-pg., 2023-12-01.

Abstract

Consider the class QSH of all non-degenerate planar quadratic differential systems possessing an invariant algebraic hyperbola. In this paper we consider a specific two-parametric family of systems in QSH possessing three pairs of infinite singularities, namely family QSH(D). For this family we have generically a presence of one couple of parallel invariant straight lines and one invariant algebraic hyperbola. Our goal is to explore the relationship among the topological bifurcation diagram, geometry of configurations of invariant algebraic curves and its type of integrability. For this study we construct the topological bifurcation diagram of configurations and phase portraits of the family QSH(D) altogether. We also study the integrability, we obtain all the distinct configurations of invariant algebraic curves, and we get all the topologically distinct phase portraits in the Poincare disc. More precisely, we prove that the family under consideration is Liouvillian integrable, there are 53 distinct configurations of invariant algebraic curves, and there exist 18 topologically distinct phase portraits. (AU)

• FAPESP's process: 19/21181-0 - New frontiers in Singularity Theory
• Grantee: Regilene Delazari dos Santos Oliveira
• Support Opportunities: Research Projects - Thematic Grants