Global dynamics and integrability of quadratic polynomial differential systems in R3 with symetry

• Grant number: 14/14096-2
• Support type: Scholarships in Brazil - Master
• Effective date (Start): September 01, 2014
• Effective date (End): August 31, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Cooperation agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
• Principal Investigator: Marcelo Messias
• Grantee: Naiara Aparecida dos Santos Silva
• Home Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil
• Associated research grant: 13/24541-0 - Ergodic and qualitative theory of dynamical systems, AP.TEM

Abstract

In the present research project we propose the study of global analysis and integrability of quadratic polynomial differential systems defined on the space R3, which has D_2 symetry. The global analysis proposed consists basicaly in three steps: 1) determination of some classes of quadratic vector fields which has D_2 symetry taking to acount some recent publications on this subject; 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 correponds to the points of R3 at infinity; 3) determination of invariant algebraic surfaces for these systems and 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 symetryc quadratic 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 and the consequent break of symetry 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 behaviour of the solutions of the studied systems. 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)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

SILVA, Naiara Aparecida dos Santos. Dinâmica e bifurcações de campos vetoriais polinomiais em R3 com um cilindro invariante. 2016. 79 f. Master's Dissertation - Universidade Estadual Paulista "Júlio de Mesquita Filho" Faculdade de Ciências e Tecnologia..