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

Grant number: 13/26602-7
Support type: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
Cooperation agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
Principal researcher:Marcelo Messias
Grantee:Alisson de Carvalho Reinol
Home 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)

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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)
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. Web of Science Citations: 0.
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, MAY 2018. Web of Science Citations: 1.
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, MAR 2018. Web of Science Citations: 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. Web of Science Citations: 0.
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, APR 2017. Web of Science Citations: 11.
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 JUL 2016. Web of Science Citations: 3.
Academic Publications
(References retrieved automatically from State of São Paulo Research Institutions)
REINOL, Alisson de Carvalho. Integrabilidade e dinâmica global de sistema diferenciais polinomiais definidos em R³ com superfícies algébricas invariantes de graus 1 e 2. 2017. 125 f. Doctoral Thesis - Universidade Estadual Paulista "Júlio de Mesquita Filho" Instituto de Biociências, Letras e Ciências Exatas..

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