Now showing items 1-13 of 13

    • 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: 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 of cubic-linear planar polynomial differential systems 

      Giné, Jaume; Valls, Claudia (Elsevier, 2016)
      For the cubic–linear polynomial planar differential systems with a finite singular point, we classify the ones which have a local analytic first integral around the origin and the ones that have a global analytic first integral.
    • Analytic reducibility of nondegenerate centers: Cherkas systems 

      Giné, Jaume; Llibre, Jaume (Bolyai Institute, University of Szeged; Hungarian Academy of Sciences, 2016)
      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. x˙=y,y˙=P0(x)+P1(x)y+P2(x)y2, We also study the ...
    • Center conditions and limits cycles for bilienard systems 

      Giné, Jaume (Texas State University, 2017)
      In this article we study the center problem for polynomial BiLiénard systems of degree n. Computing the focal values and using Gröbner bases we end the center conditions for such systems for n = 6. We also establish ...
    • Center conditions for generalized polynomial Kukles systems 

      Giné, Jaume (American Institute of Mathematical Sciences, 2017)
      In this paper we study the center problem for certain generalized Kukles systems \[ \dot{x}= y, \qquad \dot{y}= P_0(x)+ P_1(x)y+P_2(x) y^2+ P_3(x) y^3, \] where $P_i(x)$ are polynomials of degree $n$, $P_0(0)=0$ and $P_0'(0) ...
    • Center conditions for nilpotent cubic systems using Cherkas method 

      Giné, Jaume (Elsevier, 2016)
      In this work we study the center problem of a cubic polynomial differential system with nilpotent linear part. The analysis is based on the application of the Cherkas method to the Takens normal form. The study needs many ...
    • Center conditions for polynomial Liénard systems 

      Giné, Jaume (Springer International Publishing, 2017)
      In this paper we study the center problem for polynomial Liénard systems of degree n with damping of degree n. Computing the focal values we find the center conditions for such systems for n=5 and using modular arithmetics ...
    • Integrability conditions of a resonant saddle in Liénard-like complex systems 

      Giné, Jaume; Valls, Claudia (Elsevier, 2016)
      We consider a complex differential system with a resonant saddle that remind the classical Liénard systems in the real plane. For such systems we determine the conditions of analytic integrability of the resonant saddle.
    • Integrability of Liénard systems with a weak saddle 

      Gasull i Embid, Armengol; Giné, Jaume (Springer International Publishing, 2017)
      We characterize the local analytic integrability of weak saddles for complex Lienard systems, x˙ = y−F(x), y˙ = ax, 0 = a ∈ C, with F analytic at 0 and F(0) = F (0) = 0. We prove that they are locally integrable at the ...
    • 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 ...
    • Liénard equation and its generalizations 

      Giné, Jaume (World Scientific Publishing, 2017)
      In this paper, we first present a survey of the known results on limit cycles and center conditions for Liénard differential systems. Next we propose a generalization of such systems and we study their center conditions ...
    • 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, $\dot x=y-F(x),$ $\dot y= x$, 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 ...