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Global analysis of polynomial differential systems defined on the space R3


With the present research project we propose the global analysis of polynomial differential systems defined on the space R3, arising from the mathematical modeling of natural phenomena (Physics, Biology and Egineering) and also from theoretical point of view. The global analysis proposed consists basicaly in three steps: 1) determination of first integrals and invariant algebraic surfaces (Darboux polynomials); 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) 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 behaviour 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 (4)
(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.. Darboux invariants for planar polynomial differential systems having an invariant conic. ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, v. 65, n. 6, p. 1127-1136, . (13/01743-7, 12/18413-7)
SCARABELLO, MARLUCE DA CRUZ; MESSIAS, MARCELO. Bifurcations Leading to Nonlinear Oscillations in a 3D Piecewise Linear Memristor Oscillator. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 24, n. 1, . (09/11699-0, 12/18413-7)
GOUVEIA, MARCIO R. A.; MESSIAS, MARCELO; PESSOA, CLAUDIO. Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system. NONLINEAR DYNAMICS, v. 84, n. 2, p. 703-713, . (13/24541-0, 12/18413-7)
LLIBRE, JAUME; MESSIAS, MARCELO; REINOL, ALISSON DE CARVALHO. Normal Forms for Polynomial Differential Systems in R-3 Having an Invariant Quadric and a Darboux Invariant. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 25, n. 1, . (13/01743-7, 12/18413-7)

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