Quadratic vector fields defined in R3 with invariant planes

• Grant number: 16/01258-0
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: March 15, 2016
• End date: June 14, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Marcelo Messias
• Grantee: Alisson de Carvalho Reinol
• Supervisor: Jaume Llibre
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil
• Institution abroad: Universitat Autònoma de Barcelona (UAB), Spain
• Associated to the scholarship: 13/26602-7 - Integrability and global dynamics of quadratic vector fields defined on R3 with Quadrics as invariant surfaces, BP.DR

Abstract

The number of invariant straight lines that a polinomial differential system defined in R2 can have, as a function of its degree, and the realization of this number, were studied by several authors. However, with relation to the polynomial differential systems defined in R3, until now it was studied only the maximum number of invariant planes which this systems can have, but not is known about the realization of this number.It is known that quadratic polynomial differential systems defined in R3 with a finite number of invariant planes can have at most seven invarinat planes. Our aim with this project is to study if this number is or is not realizable. The problem proposed can be seen as a subproject of the PhD project which is being developed by the student, in which he is studying quadratic polynomial differential systems in R3 with invariant quadrics. (AU)

Scientific publications

(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)

LLIBRE, JAUME; MESSIAS, MARCELO; REINOL, ALISSON C.. Quadratic three-dimensional differential systems having invariant planes with total multiplicity nine. RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO, v. 67, n. 3, p. 569-580, DEC 2018. (13/26602-7, 16/01258-0, 13/24541-0)

LLIBRE, JAUME; MESSIAS, MARCELO; REINOL, ALISSON C.. Normal forms and global phase portraits of quadratic and cubic integrable vector fields having two nonconcentric circles as invariant algebraic curves. DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, v. 32, n. 3, p. 374-390, SEP 2017. (13/26602-7, 16/01258-0, 13/24541-0)