Characterization of the generalized Liénard polynomial systems having special invariant algebraic curves

• Xinjie Qian ^[1] ; Shaoqing Wang ^[2] ; Jiazhong Yang ^[3]
  1. [1] Jinling Institute of Technology

     Jinling Institute of Technology

     China

     
  2. [2] Central China Normal University

     Central China Normal University

     China

     
  3. [3] Peking University

     Peking University

     China

     

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 6, 2025
• Idioma: inglés
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• Resumen

  • The generalized Liénard polynomial systems of type (m, n) are the planar differential systems of the form x˙ = y, y˙ = − fm(x)y − gn(x), where fm(x) and gn(x) are two polynomials with real coefficients of degree m and n, respectively. We mainly present in this paper a complete characterization of all the generalized Liénard systems of type (m, n) having an irreducible invariant algebraic curve F(x, y) = s j=0 a j(x)ys− j = 0, where a0(x), a1(x), ··· , as(x) are polynomials of degree no more than s, s ≥ 4, and a0(x) = 0, (s − 1)a2 1 (x) = 2sa2(x). Additionally, in some cases, we establish conditions on the invariant algebraic curve to ensure the integrability of the generalized Liénard systems.

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