Now showing items 1-20 of 21

    • A Chebyshev criterion for Abelian integrals 

      Grau Montaña, Maite; Mañosas, Francesc; Villadelprat, Jordi (American Mathematical Society, 2010)
      We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals ...
    • A counterexample to the composition condition conjecture for polynomial Abel differential equations 

      Giné, Jaume; Grau Montaña, Maite; Santallusia Esvert, Xavier (Cambridge University Press, 2019)
      Polynomial Abel differential equations are considered a model problem for the classical Poincaré center–focus problem for planar polynomial systems of ordinary differential equations. In the last few decades, several works ...
    • A note on Liouvillian first integrals and invariant algebraic curves 

      Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (Elsevier, 2013-02)
      In this work we study the existence and non-existence of finite invariant algebraic curves for a complex planar polynomial differential system having a Liouvillian first integral.
    • A survey on the inverse integrating factor 

      García, I. A. (Isaac A.); Grau Montaña, Maite (Springer, 2010)
      The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles ...
    • Averaging theory at any order for computing periodic orbits 

      Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (Elsevier, 2013-05)
      We provide a recurrence formula for the coefficients of the powers of ε in the series expansion of the solutions around ε=0 of the perturbed first-order differential equations. Using it, we give an averaging theory at any ...
    • Composition conditions in the trigonometric Abel equation 

      Giné, Jaume; Grau Montaña, Maite; Santallusia Esvert, Xavier (Shanghai Normal University & Wilmington Scientific Publisher, 2013)
      In this paper we deal with the center problem for the trigonometricAbel equation dρ/dρ =a1(θ)ρ^2+a2(θ)ρ^3; where a1(θ) and a2(θ)are trigonometric polynomials in θ. This problem is closely connectedwith the classical Poincar´e ...
    • Divergence and Poincare-Liapunov constants for analytic differential systems 

      Grau Montaña, Maite; Llibre, Jaume (Elsevier, 2015)
      We consider a planar autonomous real analytic differential system with a monodromic singular point p. We deal with the center problem for the singular point p. Our aim is to highlight some relations between the divergence ...
    • Effective construction of Poincaré-Bendixson regions 

      Gasull, Armengol; Giacomini, Héctor; Grau Montaña, Maite (Shanghai Normal University & Wilmington Scientific Publisher, 2017)
      This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré-Bendixson regions by using transversal curves, that ...
    • Essential perturbations of polynomial vector fields with a period annulus 

      Buica, Adriana; Giné, Jaume; Grau Montaña, Maite (American Institute of Mathematical Sciences, 2015)
      Chicone-Jacobs and Iliev found the essential perturbations of quadratic systems when considering the problem of finding the cyclicity of a period annulus. Given a perturbation of a particular family of centers of polynomial ...
    • Integrability of planar polynomial differential systems through linear differential equations 

      Giacomini, Héctor; Giné, Jaume; Grau Montaña, Maite (Rocky Mountain Mathematics Consortium, 2006)
      In this work we consider rational ordinary differential equations dy/dx = Q(x, y)/P(x, y), with Q(x, y) and P(x, y) coprime polynomials with real coefficients. We give a method to construct equations of this type for ...
    • Limit cycles bifurcating from planar polynomial quasi--homogeneous 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 quasi-homogeneous center, which can be obtained using first order averaging method. ...
    • 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 non-autonomous polynomial vector fields were completed. The classical Darboux integrability theory and its recent extensions are based ...
    • 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 quasi-homogeneous polynomial differential systems 

      Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (American Institute of Mathematical Sciences, 2013-10)
      In this paper we find necessary and sufficient conditions in order that a planar quasi-homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasi-homogeneous ...
    • The center problem and composition condition for Abel differential equations 

      Giné, Jaume; Grau Montaña, Maite; Santallusia Esvert, Xavier (Elsevier, 2015)
      The classical Poincaré center-focus problem for planar polynomial systems of ordinary differential equations can be transformed, in certain particular cases, to the center problem of a trigonometric Abel differential ...
    • The inverse integrating factor and the Poincaré map 

      García, I. A. (Isaac A.); Giacomini, Héctor; Grau Montaña, Maite (American Mathematical Society, 2010)
      This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ...
    • The role of algebraic solutions in planar polynomial differential systems 

      Giacomini, Héctor; Giné, Jaume; Grau Montaña, Maite (Cambridge University Press, 2007)
      We study a planar polynomial differential system, given by . We consider a function , where gi(x) are algebraic functions of with ak(x) and algebraic functions, A0(x,y) and A1(x,y) do not share any common factor, h2(x) is ...
    • Transversal conics and the existence of limit cycles 

      Giacomini, Héctor; Grau Montaña, Maite (Elsevier, 2015)
      This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré-Bendixson regions by using transversal conics. We present ...
    • 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 ...
    • Universal centers in the cubic trigonometric Abel equation 

      Giné, Jaume; Grau Montaña, Maite; Santallusia Esvert, Xavier (Bolyai Institute. University of Szeged; Hungarian Academy of Sciences, 2014)
      We study the center problem for the trigonometric Abel equation dρ/dθ=a1(θ)ρ2+a2(θ)ρ3,dρ/dθ=a1(θ)ρ2+a2(θ)ρ3, where a1(θ)a1(θ) and a2(θ)a2(θ) are cubic trigonometric polynomials in θθ. This problem is closely connected with ...