Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

• Qian, Xinjie ^[1] ; Shen, Yang ^[2] ; Yang, Jiazhong ^[2]
  1. [1] Jinling Institute of Technology

     Jinling Institute of Technology

     China

     
  2. [2] Peking University

     Peking University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00484-8
• Enlaces
  • Texto completo
• Resumen

  • In this paper we deal with the Liénard system x˙=y,y˙=−fm(x)y−gn(x), where fm(x) and gn(x) are real polynomials of degree m and n, respectively. We call this system the Liénard system of type (m, n). For this system, we proved that if m+1≤n≤[4m+23], then the maximum number of hyperelliptic limit cycles is n−m−1, and this bound is sharp. This result indicates that the Liénard system of type (m,m+1) has no hyperelliptic limit cycles. Secondly, we present examples of irreducible algebraic curves of arbitrary high degree for Liénard systems of type (m,2m+1). Moreover, these systems have a rational first integral. Finally, we proved that the Liénard system of type (2, 5) has at most one hyperelliptic limit cycle, and this bound is sharp.

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