Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing Either a Cusp Point or Two Finite Saddle-Nodes

Author(s)

Artes, Joan C. ^[1] ; Oliveira, Regilene D. S. ^[2] ; Rezende, Alex C. ^[3]

Affiliation

^[1] Univ Autonoma Barcelona, Dept Matemat, Bellaterra - Spain

^[2] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Sao Carlos - Brazil

^[3] Univ Fed Sao Carlos, Dept Matemat, Sao Carlos - Brazil

• Document type: Journal article
• Source: Journal of Dynamics and Differential Equations; v. 33, n. 4, p. 1779-1821, DEC 2021.
• Web of Science Citations: 3

Abstract

The goal of this paper is to contribute to the classification of the phase portraits of planar quadratic differential systems according to their structural stability. Artes et al. (Mem Am Math Soc 134:639, 1998) proved that there exist 44 structurally stable topologically distinct phase portraits in the Poincare disc modulo limit cycles in this family, and Artes et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) showed the existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits of codimension-one quadratic systems, modulo limit cycles. In this work we begin the classification of planar quadratic systems of codimension two in the structural stability. Combining the sets of codimension-one quadratic vector fields one to each other, we obtain ten new sets. Here we consider setAAobtained by the coalescence of two finite singular points, yielding either a triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible topological phase portraits of setAAand prove their realization. We got 34 new topologically distinct phase portraits in the Poincare disc modulo limit cycles. Moreover, in this paper we correct a mistake made by the authors in the book of Artes et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) and we reduce to 203 the number of topologically distinct phase portrait of codimension one modulo limit cycles. (AU)

• FAPESP's process: 18/21320-7 - Investigation of planar quadratic differential systems of codimension two
• Grantee: Alex Carlucci Rezende
• Support Opportunities: Scholarships abroad - Research
• FAPESP's process: 14/00304-2 - Singularities of differentiable mappings: theory and applications
• Grantee: Maria Aparecida Soares Ruas
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 17/20854-5 - Qualitative theory of ordinary differential equations: integrability, periodic orbits and phase portraits
• Grantee: Regilene Delazari dos Santos Oliveira
• Support Opportunities: Regular Research Grants