Navegando por Autor "Oliveira, Regilene Delazari dos Santos"
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- ItemClassification of the family of quadratic differential systems possessing invariant ellipses.(2019-04) Oliveira, Regilene Delazari dos Santos; Rezende, Alex C.; Schlomiuk, Dana; Vulpe, NicolaeConsider the class QS of all non-degenerate quadratic systems. Note that each quadratic polynomial differential system can be identiffed with a point of R12 through its coeffcients. In this paper we provide necessary and suffcient conditions for a system in QS, in term of its coeffcients, to have at least one invariant ellipse. Let QSE be the whole class of non-degenerate planar quadratic differential systems possessing at least one invariant ellipse. For the class QSE, we give the global \bifurcation" diagram which indicates where an ellipse is present or absent and in case it is present, the diagram indicates if the ellipse is or not a limit cycle. The diagram is expressed in terms of affne invariant polynomials and it is done in the 12-dimensional space of parameters. This diagram is also an algorithm for determining for each quadratic system if it possesses an invariant ellipse and whether or not this ellipse is a limit cycle.
- ItemGlobal dynamics of the May-Leonard system with a Darboux invariant.(2019-01) Oliveira, Regilene Delazari dos Santos; Valls, CláudiaWe study the global dynamics of the classic May-Leonard model in R³. Such model depends on two real parameters and its global dynamics is known when the system is completely integrable. Using the Poincaré compactification on R³ we get the global dynamics of the classical May–Leonard differential system in R³ when β = −1 − α, a non-integrable case. In this case it admits a Darboux invariant. We provide the global phase portrait in each octant and in the Poincaré ball, that is, the compactification of R³ in the sphere S² at infinity. We also describe the ω-limit and α-limit of each of the orbits. For some values of the parameter α we find a separatrix cycle F formed by orbits connecting the finite singular points on the boundary of the first octant and every orbit on this octant has F as the ω-limit. The same holds for the sixth and eighth octants.
- ItemOn the limit cycle of a Belousov-Zabotinsky differential systems.(2019-01) Llibre, Jaume; Oliveira, Regilene Delazari dos SantosThe authors in  shown numerically the existence of a limit cycle surrounding the unstable node that system (1) has in the positive quadrant for specific values of the parameters. System (1) isone of the Belousov-Zabotinsky dynamical models. The objective of this paper is to prove that system (1), when in the positive quadrant Q has an unstable node or focus, has at least one limit cycle and, when f = 2/3, q = E²/2 and E > 0 suffciently small this limit cycle is unique.
- ItemQuadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariant.(2019-01) Llibre, Jaume; Oliveira, Regilene Delazari dos Santos; Rodrigues, CamilaThe planar quadratic systems having a Darboux invariant defined by invariant straight lines of total multiplicity two or by an invariant conic have been studied in  and , respectively. Here we shall present the normal forms of the planar quadratic systems having an invariant cubic. Moreover we classify the phase portraits in the Poincare disc of all planar quadratic polynomial differential systems with invariant cubic curve and having a Darboux invariant defined by it.
- ItemStructurally unstable quadratic vector fields of codimension two: families possessing either a cusp point or two finite saddle-nodes.(2019-01) Artés, Joan C.; Oliveira, Regilene Delazari dos Santos; Rezende, Alex C.The goal of this work is to contribute to the classification of the phase portraits of planar quadratic differential systems according to their structural stability. Artés, Kooij and Llibre 7 (1998) proved that there exist 44 structurally stable topologically distinct phase portraits in the 8 Poincaré disc modulo limit cycles in this family, and Artés, Llibre and Rezende (2018) showed the 9 existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits 10 of codimension-one quadratic systems, modulo limit cycles. In this work we begin the classification 11 of planar quadratic systems of codimension two in the structural stability. Combining the groups 12 of codimension-one quadratic vector fields one to each other, we obtain ten new groups. Here 13 we consider group AA obtained by the coalescence of two finite singular points, yielding either a 14 triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible 15 topological phase portraits of group AA and prove their realization. We got 34 new topologically 16 distinct phase portraits in the Poincaré disc modulo limit cycles.