Advanced search
Start date

Integrability and global dynamics of quadratic vector fields defined on R3 with Quadrics as invariant surfaces

Grant number: 13/26602-7
Support Opportunities:Scholarships in Brazil - Doctorate
Effective date (Start): June 01, 2014
Effective date (End): May 31, 2017
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Acordo de Cooperação: Coordination of Improvement of Higher Education Personnel (CAPES)
Principal Investigator:Marcelo Messias
Grantee:Alisson de Carvalho Reinol
Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
Associated scholarship(s):16/01258-0 - Quadratic vector fields defined in R3 with invariant planes, BE.EP.DR


With the present research project we propose the global analysis of polynomial differential systems defined on the space R3, which has a quadric as an invariant surface. The global analysis proposed consists basicaly in three steps: 1) determination of the quadratic vector fields which has a quadric as an invariant algebraic surfaces; 2) Poincaré compactification of the systems, which enables their extension to analytic systems defined on the closed ball of radius one (Poincaré ball), whose boundary, the sphere S2 (Poincaré sphere) is invariant under the flow of the extended system and corresponds to the points of R3 at infinity; 3) study of the solutions on the invariant algebraic surfaces and how this surfaces are contained in the Poincaré ball; study of the end of these surfaces at infinity (intersection with the Poincaré sphere) and consequently the description of the dynamics at infinity. The proposed analysis enable us to describe important global structures of the polynomial systems on the whole phase space R3. Furthermore, an analytical/numerical study shows that small perturbations of these global structures by varying the parameters of the systems may lead to the creation of chaotic dynamics. In this way, the understanding of such structures is an important starting point to the understanding of the complex dynamical behavior of the solutions of polynomial systems on R3. In the proposed analysis we will use the classical results of the qualitative theory and bifurcations of ordinary differential equations, combined with numerical simulations through the software MAPLE. (AU)

News published in Agência FAPESP Newsletter about the scholarship:
Articles published in other media outlets (0 total):
More itemsLess items

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)
MESSIAS, MARCELO; REINOL, ALISSON C.. On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nos,-Hoover oscillator. NONLINEAR DYNAMICS, v. 92, n. 3, p. 1287-1297, . (13/24541-0, 13/26602-7)
MESSIAS, MARCELO; REINOL, ALISSON C.. On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system. NONLINEAR DYNAMICS, v. 88, n. 2, p. 807-821, . (13/24541-0, 13/26602-7)
DALBELO, THAIS MARIA; MESSIAS, MARCELO; REINOL, ALISSON C.. Polynomial Differential Systems in Having Invariant Weighted Homogeneous Surfaces. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 49, n. 1, p. 137-157, . (13/26602-7, 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, . (13/26602-7, 16/01258-0, 13/24541-0)
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, . (13/26602-7, 16/01258-0, 13/24541-0)
MESSIAS, MARCELO; REINOL, ALISSON C.. Integrability and Dynamics of Quadratic Three-Dimensional Differential Systems Having an Invariant Paraboloid. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 26, n. 8, . (13/24541-0, 13/26602-7)
Academic Publications
(References retrieved automatically from State of São Paulo Research Institutions)
REINOL, Alisson de Carvalho. Integrability and global dynamics of polynomial differential systems defined in R³ with invariant algebraic surfaces of degrees 1 and 2. 2017. Doctoral Thesis - Universidade Estadual Paulista (Unesp). Instituto de Biociências Letras e Ciências Exatas. São José do Rio Preto São José do Rio Preto.

Please report errors in scientific publications list by writing to: