Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

Abstract

In this paper we deal with the Liénard system $\dot{x}=y, \dot{y}=-f_m(x)y-g_n(x),$ where $f_m(x)$ and $g_n(x)$ 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 $m+1\le n\le [\frac{4m+2}{3}]$, 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.