Browsing Articles publicats (Matemàtica) by Author "Llibre, Jaume"
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Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems
Giné, Jaume; Llibre, Jaume (MDPI, 2022)In this study, we focus on invariant algebraic curves of generalized Liénard polynomial differential systems x 0 = y, y 0 = −fm(x)y − gn(x), where the degrees of the polynomials f and g are m and n, respectively, and ... 
Limit cycles bifurcating from planar polynomial quasihomogeneous centers
Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (Elsevier, 2015)In this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasihomogeneous center, which can be obtained using first order averaging method. ... 
On the center conditions for analytic monodromic degenerate singularities
Giné, Jaume; Llibre, Jaume (World Scientific Publishing, 2012)In this paper we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields can be applied independently that the ... 
On the extensions of the Darboux theory of integrability
Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (IOP Publishing, 2013)Recently some extensions of the classical Darboux integrability theory to autonomous and nonautonomous polynomial vector fields were completed. The classical Darboux integrability theory and its recent extensions are based ... 
On the Integrability of Liénard systems with a strong saddle
Giné, Jaume; Llibre, Jaume (Elsevier, 2017)We study the local analytic integrability for real Li\'{e}nard systems, $\dot x=yF(x),$ $\dot y= x$, with $F(0)=0$ but $F'(0)\ne0,$ which implies that it has a strong saddle at the origin. First we prove that this problem ... 
On the multiple zeros of a real analytic function with applications to the averaging theory of differential equations
García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (IOP Publishing, 20181121)In this work we consider real analytic functions $d(z,\la,\e)$, where $d : \Omega \times \mathbb{R}^p \times I \to \Omega$, $\Omega$ is a bounded open subset of $\R$, $I \subset \mathbb{R}$ is an interval containing the ... 
On the periodic orbit bifurcating from a Hopf bifurcation in systems with two slow and one fast variables
García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Elsevier, 2014)The Hopf bifurcation in slowfast systems with two slow variables and one fast variable has been studied recently, mainly from a numerical point of view. Our goal is to provide an analytic proof of the existence of the ... 
On the planar integrable differential systems
Giné, Jaume; Llibre, Jaume (Springer Verlag, 2011)Under very general assumptions we prove that the planar differential systems having a first integral are essentially the linear differential systems u˙ = u, ˙v = v. Additionally such systems always have a Lie symmetry. We ... 
Periodic solutions for nonlinear differential systems: the second order bifurcation function
Buica, Adriana; Giné, Jaume; Llibre, Jaume (Nicolaus Copernicus University in Torun, Juliusz Schauder Centre for Nonlinear Studies, 2014)We are concerned here with the classical problem of Poincaré of persistence of periodic solutions under small perturbations. The main contribution of this work is to give the expression of the second order bifurcation ... 
Polynomial and rational first integrals for planar homogeneous polynomial differential systems
Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (Universitat Autònoma de Barcelona, 2014)In this paper we find necessary and suficient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic ... 
Polynomial and rational first integrals for planar quasihomogeneous polynomial differential systems
Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (American Institute of Mathematical Sciences, 201310)In this paper we find necessary and sufficient conditions in order that a planar quasihomogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasihomogeneous ... 
Reversible nilpotent centers with cubic homogeneous nonlinearities
Dukaric, Masa; Giné, Jaume; Llibre, Jaume (Elsevier, 2016)We provide 13 nontopological 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 ... 
Strongly formal Weierstrass nonintegrability for polynomial differential systems in C2
Giné, Jaume; Llibre, Jaume (Bolyai Institute, University of Szeged; Hungarian Academy of Sciences, 2020)Recently a criterion has been given for determining the weakly formal Weierstrass nonintegrability of polynomial differential systems in C2 . Here we extend this criterion for determining the strongly formal Weierstrass ... 
The center problem for Z_2symmetric nilpotent vector fields
Algaba, Antonio; García, Cristóbal; Giné, Jaume; Llibre, Jaume (Elsevier, 2018)We say that a polynomial differential system ˙x = P(x, y), ˙y = Q(x, y) having the origin as a singular point is Z2symmetric if P(−x, −y) = −P(x, y) and Q(−x, −y) = −Q(x, y). It is known that there are nilpotent centers ... 
The cubic polynomial differential systems with two circles as algebraic limit cycles
Giné, Jaume; Llibre, Jaume; Valls, Claudia (Walter de Gruyter GmbH, 2018)In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles. 
The Hopf cyclicity of the centers of a class of quintic polynomial vector fields
García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Elsevier, 20150120)We consider families of planar polynomial vector fields having a singularity with purely imaginary eigenvalues for which a basis of its Bautin ideal B is known. We provide an algorithm for computing an upper bound of the ... 
The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems
Giacomini, Héctor; Giné, Jaume; Llibre, Jaume (Elsevier, 2006)In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré–Liapunov ... 
Universal centers and composition conditions
Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (London Mathematical Society, 2013)In this paper, we characterize the universal centres of the ordinary differential equations , where ai(θ) are trigonometric polynomials, in terms of the composition conditions. These centres are closely related with the ... 
Vanishing set of inverse Jacobi multipliers and attractor/repeller sets
García, I. A. (Isaac A.); Giné, Jaume; Llibre, Jaume; Maza Sabido, Susanna (American Institute of Physics, 20210101)In this paper we study conditions under which the zeroset of the inverse Jacobi multiplier of a smooth vector field contains its attractor/repeller compact sets. The work generalizes previous results focusing on sink ... 
Weierstrass integrability in Liénard differential systems
Giné, Jaume; Llibre, Jaume (Elsevier, 2011)In this work we study the Liénard differential systems that admit a Weierstrass first integral or a Weierstrass inverse integrating factor