Now showing items 1-12 of 12

    • A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields 

      Algaba, Antonio; García, Cristóbal; Giné, Jaume (Springer, 2021)
      In this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In ...
    • Analytic integrability around a nilpotent singularity 

      Algaba, Antonio; García, Cristóbal; Giné, Jaume (Elsevier, 2019)
      In this work it is characterized the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions.
    • Analytic integrability around a nilpotent singularity: The non-generic case 

      Algaba, Antonio; Díaz, María; García, Cristóbal; Giné, Jaume (American Institute of Mathematical Sciences, 2020)
      Recently, in [9] is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. In this work we solve the remaining case completing the ...
    • Analytic integrability inside a family of degenerate centers 

      Algaba, Antonio; Checa, Isabel; García, Cristóbal; Giné, Jaume (Elsevier, 2016)
      In this paper we study the analytic integrability around the origin inside a family of degenerate centers or perturbations of them. For this family analytic integrability does not imply formal orbital equivalence to a ...
    • Analytic integrability of some examples of degenerate planar vector fields 

      Algaba, Antonio; García, Cristóbal; Giné, Jaume (Springer, 2016)
      This paper is devoted to the classification of analytic integrable cases of two families of degenerate planar vector fields with a monodromic singular point at the origin. This study falls in the still open degenerate ...
    • Center problem for generic degenerate vector fields 

      Algaba, Antonio; Díaz, María; García, Cristóbal; Giné, Jaume (Elsevier, 2022)
      We generalize the method of construction of an integrating factor for Abel differential equations, developed in Briskin et al. (1998), for any generic monodromic singularity. Here generic means that the vector field has ...
    • Geometric criterium in the center problem 

      Algaba, Antonio; García, Cristóbal; Giné, Jaume (Springer Basel, 2016-10)
      In this paper we use a geometric criterium based in the classical method of the construction of Lyapunov functions to determine if a differential system has a focus or a center at a singular point. This criterium is proved ...
    • Integrability of planar nilpotent differential systems through the existence of an inverse integrating factor 

      Algaba, Antonio; García, Cristóbal; Giné, Jaume (Elsevier, 2019)
      In this work is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. The analytic integrability problem is characterized via the ...
    • Nilpotent centres via inverse integrating factors 

      Algaba, Antonio; García, Cristóbal; Giné, Jaume (Cambridge University Press, 2016-10)
      In this paper we are interested in the nilpotent center problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centers. More general objects are ...
    • On the Formal Integrability Problem for Planar Differential Systems 

      Giné, Jaume; Algaba, Antonio; García, Cristóbal (Hindawi Publishing Corporation, 2013)
      We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results ...
    • Orbital Reversibility of Planar Vector Fields 

      Algaba, Antonio; García, Cristóbal; Giné, Jaume (MDPI, 2021)
      In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility ...
    • The center problem for Z_2-symmetric 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 Z2-symmetric if P(−x, −y) = −P(x, y) and Q(−x, −y) = −Q(x, y). It is known that there are nilpotent centers ...