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1.
15 p, 365.0 KB Quadratic systems with an invariant conic having Darboux invariants / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Oliveira, Regilene D. S. (Universidade de Sâo Paulo (Brazil). Departamento de Matemática)
The complete characterization of the phase portraits of real planar quadratic vector fields is very far to be completed. As this attempt is not possible in the whole class due to the large number of parameters (twelve, but, after affine transformations and time rescaling, we arrive at families with five parameters, which is still a big number of parameters), many subclasses have been considered and studied. [...]
2018 - 10.1142/S021919971750033X
Communications in contemporaray mathematics, Vol. 20, issue 4 (2018) , art. 1750033  
2.
43 p, 1.7 MB Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Schlomiuk, Dana (Université de Montréal. Département de Mathématiques et de Statistiques) ; Vulpe, Nicolae (Academy of Science of Moldova. Institute of Mathematics and Computer Science)
In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. [...]
2013 - 10.1007/s11784-014-0175-2
Journal of Fixed Point Theory and Applications, Vol. 14 Núm. 2 (2013) , p. 555-618  
3.
15 p, 739.0 KB Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia (Universidade Técnica de Lisboa. Departamento de Matemática)
We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse. More precisely, a quadratic system having an invariant ellipse can be written into the form ˙x = x2+y2−1+y(ax+by+c), ˙y = −x(ax+by+c), and the ellipse becomes x2+y2 = 1. [...]
2014 - 10.1007/s10114-014-2484-1
Acta Mathematica Sinica. English Series, Vol. 30 Núm. 3 (2014) , p. 453-466  
4.
33 p, 583.3 KB The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (A,B) / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Rezende, Alex C. (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. [...]
2014 - 10.1142/S0218127414500448
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Vol. 24 Núm. 4 (2014) , p. 1450044 (30 pages)  
5.
62 p, 2.4 MB Geometric configurations of singularities for quadratic differential systems with total finite multiplicity m_f=2 / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Schlomiuk, Dana (Université de Montréal. Département de Mathématiques et de Statistiques) ; Vulpe, Nicolae (Academy of Science of Moldova. Institute of Mathematics and Computer Science)
In this work we consider the problem of classifying all configurations of singularities, both finite and infinite of quadratic differential systems, with respect to the geometric equivalence relation defined in [3]. [...]
2014
Electronic Journal of Differential Equations, Vol. 2014 Núm. 159 (2014) , p. 1-79  
6.
12 p, 370.0 KB Piecewise bounded quadratic systems in the plane / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; da Silva, Clayton Eduardo Lente (UNESP(Brazil). Deparatamento de Matemática) ; da Silva, Paulo Ricardo (UNESP(Brazil). Deparatamento de Matemática)
In this paper we study the sliding mode of piecewise bounded quadratic systems in the plane given by a non-smooth vector field Z = (X, Y ). Analyzing the singular, crossing and sliding sets, we get the conditions which ensure that any solution, including the sliding one, is bounded.
2014 - 10.1007/s12591-014-0205-y
Differential Equations and Dynamical Systems, 2014  
7.
35 p, 1.6 MB Global configurations of singularities for quadratic differential systems with total finite multiplicity three and at most two real singularities / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Schlomiuk, Dana (Université de Montréal. Département de Mathématiques et de Statistiques) ; Vulpe, Nicolae (Academy of Science of Moldova. Institute of Mathematics and Computer Science)
In this work we consider the problem of classifying all configurations of singularities, finite and infinite, of quadratic differential systems, with respect to the geometric equivalence relation defined in [2]. [...]
2014 - 10.1007/s12346-014-0119-7
Qualitative Theory of Dynamical Systems, Vol. 13 (2014) , p. 305-351  
8.
36 p, 1.5 MB Global configurations of singularities for quadratic differential systems with exactly two finite singularities of total multiplicity four / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Rezende, Alex C. (Universidade de São Paulo) ; Schlomiuk, Dana (Université de Montréal) ; Vulpe, Nicolae (Academy of Science of Moldova)
In this work we consider the problem of classifying all configurations of singularities, both finite and infinite of quadratic differential systems, with respect to the geometric equivalence relation defined in [2]. [...]
2014 - 10.14232/ejqtde.2014.1.60
Electronic Journal of Qualitative Theory of Differential Equations, Vol. 60 (2014) , p. 1-43  
9.
40 p, 1.5 MB Geometric configurations of singularities for quadratic differential systems with total finite multiplicity lower than 2 / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Schlomiuk, Dana (Université de Montréal. Département de Mathématiques et de Statistiques) ; Vulpe, Nicolae (Academy of Science of Moldova. Institute of Mathematics and Computer Science)
In [3] we classified globally the configurations of singularities at infinity of quadratic differential systems, with respect to the geometric equivalence relation. The global classification of configurations of finite singularities was done in [2] modulo the coarser topological equivalence relation for which no distinctions are made between a focus and a node and neither are they made between a strong and a weak focus or between foci of different orders. [...]
2013
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica., Vol. 71 Núm. 1 (2013) , p. 72-124  
10.
28 p, 571.0 KB Complete geometric invariant study of two classes of quadratic systems / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Vulpe, Nicolae (Academy of Science of Moldova. Institute of Mathematics and Computer Science)
In this paper we give a complete study using affine invariant conditions for the quadratic systems having centers. Independently we do the same for quadratic systems being Hamiltonian. There are two improvements of the previous results as [29] in which centers where study up to GL-invariant, and of [1] in which Hamiltonian QS where classified without invariants. [...]
2012
Electronic Journal of Differential Equations, Vol. 2012 Núm. 9 (2012) , p. 1-35  

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