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21.
5 p, 574.4 KB Integrability conditions of a resonant saddle in generalized Liénard-like complex polynomial differential systems / Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We consider a complex differential system with a resonant saddle at the origin. We compute the resonant saddle quantities and using Gröbner bases we find the integrability conditions for such systems up to a certain degree. [...]
2017 - 10.1016/j.chaos.2017.01.014
Chaos, solitons and fractals, Vol. 96 (2017) , p. 130-131  
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22.
7 p, 267.6 KB Integrability of Liénard systems with a weak saddle / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica)
We characterize the local analytic integrability of weak saddles for complex Liénard systems.
2017 - 10.1007/s00033-016-0756-6
ZAMP. Journal of Applied Mathematics and Physics, Vol. 68 Núm. 13 (2017)  
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23.
15 p, 651.1 KB Centers of weight-homogeneous polynomial vector fields on the plane / Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Valls, Clàudia 1973- (Universidade de Lisboa. Departamento de Matemàtica)
We characterize all centers of a planar weight-homogeneous polynomial vector fields. Moreover we classify all centers of a planar weight-homogeneous polynomial vector fields of degrees 6 and 7.
2017 - 10.1090/proc/13393
Proceedings of the American Mathematical Society, Vol. 145 Núm. 6 (2017) , p. 2539-2555  
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24.
12 p, 363.9 KB Analytic reducibility of nondegenerate centers: Cherkas systems / Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
In this paper we study the center problem for polynomial differential systems and we prove that any center of an analytic differential system is analytically reducible. We also study the centers for the Cherkas polynomial differential systems x˙ = y, y˙ = P0(x) + P1(x)y + P2(x)y2, where Pi(x) are polynomials of degree n, P0(0) = 0 and P'0(0) < 0. [...]
2016 - 10.14232/ejqtde.2016.1.49
Electronic Journal of Qualitative Theory of Differential Equations, Vol. 49 (2016) , p. 1-10  
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25.
30 p, 812.6 KB Averaging methods of arbitrary order, periodic solutions and integrability / Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Wu, Kesheng (Shanghai Jiao Tong University. Department of Mathematics) ; Zhang, Xiang (Shanghai Jiao Tong University. Department of Mathematics)
In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods. [...]
2016 - 10.1016/j.jde.2015.11.005
Journal of differential equations, Vol. 260 (2016) , p. 4130-4156  
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26.
11 p, 694.7 KB A sufficient condition in order that the Real Jacobian Conjecture in R^2 holds / Braun, Francisco (Universidade Federal de Sâo Carlos Rod(Brasil). Departamento de Matemática) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Let F=(f,g):\R^2\R^2 be a polynomial map such that DF(x) is different from zero for all x\R^2 and F(0,0) = (0,0). We prove that for the injectivity of F it is sufficient to assume that the higher homogeneous terms of the polynomials ff_x g g_x and f f_y g g_y do not have real linear factors in common.
2016 - 10.1016/j.jde.2015.12.011
Journal of differential equations, Vol. 260 (2016) , p. 5250-5258  
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27.
24 p, 13.1 MB Reversible nilpotent centers with cubic homogeneous nonlinearities / Dukarić, Maša (University of Maribor(Slovenia). Center for Applied Mathematics and Theoretical Physics) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We provide 13 non-topological equivalent classes of global phase portraits in the Poincaré disk of reversible cubic homogeneous systems with a nilpotent center at origin, which complete the classification of the phase portraits of the nilpotent centers with cubic homogeneous nonlinearities.
2016 - 10.1016/j.jmaa.2015.07.049
Journal of mathematical analysis and applications, Vol. 433 (2016) , p. 305-319  
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28.
20 p, 414.4 KB Center problem for systems with two monomial nonlinearities / Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. [...]
2016 - 10.3934/cpaa.2016.15.577
Communications on pure & applied analysis, Vol. 15 Núm. 2 (2016) , p. 577-598  
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29.
6 p, 556.2 KB Analytic nilpotent centers as limits of nondegenerate centers revisited / García, Isaac (Universitat de Lleida. Departament de Matemàtica) ; Giacomini, Hector (Université de Tours(France). Laboratoire de Mathématiques et Physique Théorique) ; Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré-Liapunov method to detect centers with purely imaginary eigenvalues can be used to detect nilpotent centers.
2016 - 10.1016/j.jmaa.2016.04.046
Journal of mathematical analysis and applications, Vol. 441 (2016) , p. 893-899  
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30.
14 p, 580.6 KB A method for characterizing nilpotent centers / Giné, Jaume (Universitat de Lleida. Departament de Matemàtica) ; Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
To characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest, first for the problem of distinguishing between a focus and a center, and after for studying the bifurcation of limit cycles from it or from its period annulus. [...]
2014 - 10.1016/j.jmaa.2013.12.013
Journal of mathematical analysis and applications, Vol. 413 (2014) , p. 537-545  
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54 Gine, Jaume
54 Gine, Jaume
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