Investigation of planar quadratic differential systems of codimension two

• Grant number: 18/21320-7
• Support Opportunities: Scholarships abroad - Research
• Start date: December 09, 2018
• End date: January 31, 2019
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Alex Carlucci Rezende
• Grantee: Alex Carlucci Rezende
• Host Investigator: Joan Carles Artes
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Institution abroad: Universitat Autònoma de Barcelona (UAB), Spain

Abstract

In this research project we present the problem in which we may work during the scientific visit at the Department of Mathematics of the Autonomous University of Barcelona (UAB), in collaboration with Prof. Dr. Joan Carles Artés, and also Prof. Dr. Jaume Llibre and Prof. Dr. Joan Torregrosa. The project consists of the the investigation of planar quadratic differential systems according to their structural stability. In 1998, Artés, Kooij and Llibre proved that planar quadratic differential systems have 44 topologically distinct structurally stable phase portraits in the Poincaré disc, modulo limit cycles. As a continuation, in 2018, Artés, Llibre and Rezende showed that such systems possess 204 topologically distinct structurally unstable phase portraits of codimension one (modulo limit cycles). In the present project we begin the classification of the quadratic systems of codimension two according to their structural stability. This class splits into several subclasses and we intend to analyze them strictly in order to contribute to the global classification of the planar quadratic systems. By now, we have analyzed the subfamilies possessing two finite singular points of the type saddle-node and possessing a singular point of cusp type, and we have obtained 34 (19 plus 15) topologically distinct phase portraits.

Scientific publications

(6)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

ARTES, JOAN C.; OLIVEIRA, REGILENE D. S.; REZENDE, ALEX C.. Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing Either a Cusp Point or Two Finite Saddle-Nodes. Journal of Dynamics and Differential Equations, JUL 2020. (17/20854-5, 18/21320-7, 14/00304-2)

ARTES, JOAN C.; OLIVEIRA, REGILENE D. S.; REZENDE, ALEX C.. Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing Either a Cusp Point or Two Finite Saddle-Nodes. Journal of Dynamics and Differential Equations, v. 33, n. 4, p. 1779-1821, DEC 2021. (18/21320-7, 14/00304-2, 17/20854-5)

ARTES, JOAN C.; MOTA, MARCOS C.; REZENDE, ALEX C.. Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1,1)SN - (A). INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 31, n. 2, FEB 2021. (18/21320-7)

ARTES, JOAN C.; MOTA, MARCOS C.; REZENDE, ALEX C.. Structurally unstable quadratic vector fields of codimension two: families possessing a finite saddle-node and an infinite saddle-node. Electronic Journal of Qualitative Theory of Differential Equations, n. 35, p. 1-89, 2021. (18/21320-7, 19/21181-0)

ARTES, JOAN C.; MOTA, MARCOS C.; REZENDE, ALEX C.. Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1,1) SN - (B). INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 31, n. 09, JUL 2021. (18/21320-7, 19/21181-0)

ARTES, JOAN C.; MOTA, MARCOS C.; REZENDE, ALEX C.. Structurally unstable quadratic vector fields of codimension two: families possessing a finite saddle-node and an infinite saddle-node. Electronic Journal of Qualitative Theory of Differential Equations, v. N/A, n. 35, p. 89-pg., 2021-01-01. (19/21181-0, 18/21320-7)