Quadratic systems with an invariant conic having Darboux invariants

Author(s)

Llibre, Jaume ^[1] ; Oliveira, Regilene ^[2]

Affiliation

^[1] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona, Catalonia - Spain

^[2] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Dept Matemat, CP 668, BR-13566590 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: COMMUNICATIONS IN CONTEMPORARY MATHEMATICS; v. 20, n. 4 JUN 2018.
• Web of Science Citations: 0

Abstract

The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real aline transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these normal forms, we complete the characterization of the phase portraits in the Poincare disc of all planar quadratic polynomial differential systems having an invariant conic C: f (x, y) = 0, and a Darboux invariant of the form f (x, y)e(st) with s is an element of R\textbackslash{}[0]. (AU)

• FAPESP's process: 14/00304-2 - Singularities of differentiable mappings: theory and applications
• Grantee: Maria Aparecida Soares Ruas
• Support Opportunities: Research Projects - Thematic Grants