Navegando por Autor "Rezende, Alex C."

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    Classification of the family of quadratic differential systems possessing invariant ellipses.
    (2019-04) Oliveira, Regilene Delazari dos Santos; Rezende, Alex C.; Schlomiuk, Dana; Vulpe, Nicolae
    Consider 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.
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    Family of quadratic differential systems with irreducible invariant hyperbolas: a complete classification in the space R 12.
    (2014) Oliveira, Regilene Delazari dos Santos; Rezende, Alex C.; Vulpe, Nicolae
    In this article we consider the class QSf of all quadratic systems possessing a finite number of singularities (finite and infinite).
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    Geometric analysis of quadratic differential systems with invariant ellipses.
    (2019-10) Mota, Marcos C.; Oliveira, Regilene Delazari dos Santos; Rezende, Alex C.; Schlomiuk, Dana; Vulpe, Nicolae
    n this article we study the whole class QSE of non-degenerate planar quadratic differential systems possessing at least one invariant ellipse.We classify this family of systems according to their geometric properties encoded in the configurations of invariant ellipses and invariant straight lines which these systems could possess. The classification, which staken modulo the action of t he group of real affine transformations and time rescaling, is given in terms of algebraic geometric invariants and also in terms of invariant polynomials and it yields a total of 35 distinct such configurations. This classification is also an algorithm which makes it possible to verify for any given real quadratic differential system if it has invariant ellipses or not and to specify its configuration of invariant ellipses and straight lines.
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    Geometric and algebraic classification of quadratic differential systems with invariant hyperbolas.
    (2016-11) Oliveira, Regilene Delazari dos Santos; Rezende, Alex C.; Schlomiuk, Dana
    Let QSH be the whole class of non-degenerate planar quadratic differential systems possessing at least one invariant hyperbola.
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    Global phase portraits of a SIS model.
    (2012) Oliveira, Regilene Delazari dos Santos; Rezende, Alex C.
    In the qualitative theory of ordinary differential equations, we can find many papers whose objective is the classification of all the possible topological phase portraits of a given family of differential system.
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    Global phase portraits of quadratic polynomial differential systems with a semi-elemental triple node.
    (2013) Artés, Joan C.; Rezende, Alex C.; Oliveira, Regilene Delazari dos Santos
    Planar quadratic differential systems occur in many areas of applied mathematics.
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    Structurally 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.
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    The geometry of quadratic polynomial differential systems with a finite and an infinite Saddle-Node (C).
    (2014) Artés, Joan; Rezende, Alex C.; Oliveira, Regilene Delazari dos Santos
    Planar quadratic differential systems occur in many areas of applied mathematics.