Now showing items 1-20 of 22

    • 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 ...
    • 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 around the origin of certain differential system 

      Giné, Jaume; Valls, Claudia (Wilmington Scientific Publisher, 2022)
      In this work we consider the polynomial differential system x˙=−y+xyn−1, y˙=x+ayxn−1, where a∈R and n≥2 with n∈N. This system is a certain generalization of the classical Liénard system. We study the center problem and ...
    • 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 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 ...
    • 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 ...
    • Center problem for trigonometric Liénard systems 

      Gasull, Armengol; Giné, Jaume; Valls, Claudia (Elsevier, 2017)
      We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Liénard systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems ...
    • 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 ...
    • Integrability of complex planar systems with homogeneous nonlinearities 

      Fercec, Brigita; Giné, Jaume; Romanovski, Valery G.; Edneral, Victor F. (Elsevier, 2016)
      In this paper we obtain sufficient conditions for the existence of a local analytic first integral for a family of quintic systems having homogeneous nonlinearities. The family studied in this work is the largest one ...
    • 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 ...
    • Lie symmetries and the center problem 

      Giné, Jaume (Shanghai Normal University & Wilmington Scientific Publisher, 2011)
      Abstract In this short survey we discuss the narrow relation between the center problem and the Lie symmetries. It is well known that an analytic vector field X having a non–degenerate center has a non–trivial analytic Lie ...
    • 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 ...
    • Non-smooth quadratic centers defined in two arbitrary sectors 

      García, I. A. (Isaac A.) (Elsevier, 2016-11-25)
      In this paper we analyze the center-focus problem of some families of piecewise planar quadratic vector fields on two zones of R2. The zones we consider are two unbounded sectors defined by an arbitrary angle α and a fixed ...
    • Nondegenerate centers for Abel polynomial differential equations of second kind 

      Giné, Jaume; Valls, Claudia (Elsevier, 2017)
      In this paper we study the center problem for Abel polynomial differential equations of second kind. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems ...
    • On the Dynamics of Higgins–Selkov, Selkov and Brusellator Oscillators 

      Giné, Jaume (MDPI, 2022)
      A complete algebraic characterization of the first integrals of the Higgins–Selkov, Selkov and Brusellator oscillators is given here. The existence of symmetries sometimes forces the existence of such first integrals. ...
    • 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 ...
    • 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 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 ...