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1.
Phase portraits of uniform isochronous centers with homogeneous nonlinearities / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática (Portugal))
We classify the phase portraits in the Poincaré disc of the differential equations of the form x' = − y + xf(x, y), ẏ = x + yf(x, y) where f(x,y) is a homogeneous polynomial of degree n − 1 when n = 2, 3, 4, 5, and f has only simple zeroes. [...]
2021 - 10.1007/s10883-021-09529-2
Journal of Dynamical and Control Systems, (February 2021)  
2.
Lower bounds for the local cyclicity for families of centers / Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Gouveia, Luiz F. S (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In this paper we are interested on how the local cyclicity of a family of centers depends on the parameters. This fact, was pointed out in [21], to prove that there exists a family of cubic centers, labeled by CD12 31 in [25], with more local cyclicity than expected. [...]
2021 - 10.1016/j.jde.2020.11.035
Journal of differential equations, Vol. 275 (February 2021) , p. 309-331  
3.
Lower bounds for the local cyclicity of centers using high order developments and parallelization / Gouveia, Luiz F. S (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We are interested in small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point, that we locate at the origin. We develop a parallelization technique to study higher order developments, with respect to the parameters, of the return map near the origin. [...]
2021 - 10.1016/j.jde.2020.08.027
Journal of differential equations, Vol. 271 (January 2021) , p. 447-479  
4.
11 p, 320.0 KB A new algorithm for finding rational first integrals of polynomial vector fields / Ferragut, Antoni (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giacomini, Héctor (Centre National de la Recherche Scientifique (França))
We present a new method to compute rational first integrals of planar polynomial vector fields. The algorithm is in general much faster than the usual methods and also allows to compute the remarkable curves associated to the rational first integral of the system.
2010 - 10.1007/s12346-010-0021-x
Qualitative Theory of Dynamical Systems, Vol. 9, Issue 1-2 (November 2010) , p. 89-99  
5.
14 p, 412.2 KB Topological classification of polynomial complex differential equations with all the critical points of centre type / Álvarez Torres, María Jesús (Universitat de les Illes Balears. Departament de Matemàtiques i Informàtica) ; Gasull i Embid, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Prohens, Rafel (Universitat de les Illes Balears. Departament de Matemàtiques i Informàtica)
In this paper we study the global phase portrait of complex polynomial differential equations of degree n of the form z˙ = f(z), having all their critical points of center type. We give the exact number of topologically different phase portraits on the Poincaré disk when n ≤ 6 and, in the remaining cases, an upper bound for this number which only depends on n.
2010 - 10.1080/10236190903232654
Journal of Difference Equations and Applications, Vol. 16, Issue 5-6 (May 2010) , p. 411-423  
6.
On the computation of Darboux first integrals of a class of planar polynomial vector fields / Ferragut, Antoni (Universitat Jaume I. Institut Universitari de Matemàtiques i Aplicacions de Castelló) ; Galindo, Carlos (Universitat Jaume I. Institut Universitari de Matemàtiques i Aplicacions de Castelló) ; Monserrat, Francisco (Universitat Politècnica de València. Institut Universitari de Matemàtica Pura i Aplicada)
We study the class of planar polynomial vector fields admitting Darboux first integrals of the type ∏ri = 1fαii, where the αi's are positive real numbers and the fi's are polynomials defining curves with only one place at infinity. [...]
2019 - 10.1016/j.jmaa.2019.05.052
Journal of mathematical analysis and applications, Vol. 478, Issue 2 (October 2019) , p. 743-763  
7.
13 p, 721.0 KB Algebraic limit cycles on quadratic polynomial differential systems / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Departamento de Matemàtica)
Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: Quadratic polynomial differential systems have at most one algebraic limit cycle. [...]
2018 - 10.1017/S0013091517000244
Proceedings of the Edinburgh Mathematical Society, Vol. 61, issue 2 (May 2018) , p. 499-512  
8.
13 p, 709.8 KB Algebraic limit cycles for quadratic polynomial differential systems / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Departamento de Matemàtica)
We prove that for a quadratic polynomial differential system having three pairs of diametrally opposite equilibrium points at infinity that are positively rationally independent, has at most one algebraic limit cycle. [...]
2018 - 10.3934/dcdsb.2018070
Discrete and continuous dynamical systems. Series B, Vol. 23, issue 6 (2018) , p. 2475-2485  
9.
17 p, 722.0 KB On the uniqueness of algebraic limit cycles for quadratic polynomial differential systems with two pairs of equilibrium points at infinity / Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Departamento de Matematica)
Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and few years later the following conjecture appeared: Quadratic polynomial differential systems have at most one algebraic limit cycle. [...]
2017 - 10.1007/s10711-017-0244-y
Geometriae Dedicata, Vol. 191 (2017) , p. 37-52  
10.
8 p, 341.6 KB Uniform isochronous cubic and quartic centers: Revisited / Artés, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Itikawa, Jackson (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In this paper we completed the classification of the phase portraits in the Poincaré disc of uniform isochronous cubic and quartic centers previously studied by several authors. There are three and fourteen different topological phase portraits for the uniform isochronous cubic and quartic centers respectively.
2017 - 10.1016/j.cam.2016.09.018
Journal of computational and applied mathematics, Vol. 313 (2017) , p. 448-453  

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