Resumen de Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems
Xinjie Qian, Yang Shen, Jiazhong Yang
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.
