In this paper, we present a systematic study of the maximum number, denoted by H(m, n), of hyperelliptic limit cycles of the Liénard systems$$\begin{aligned} \dot{x}=y, \quad \dot{y}=-f_m(x)y-g_n(x), \end{aligned}$$where, respectively, \(f_m(x)\) and \(g_n(x)\) are real polynomials of degree m and n. The main results of the paper are as follows: We give the upper as well as the lower bounds of H(m, n) in all the cases. It turns out that in most cases these bounds are sharp. Furthermore, the configuration of hyperelliptic limit cycles is also explicitly described.

中文翻译：

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Liénard系统的超椭圆极限环数

在本文中，我们对Liénard系统的超椭圆极限环的最大数目（用H（m，  n）表示）进行了系统研究$$ \ begin {aligned} \ dot {x} = y，\ quad \ dot {y} =-f_m（x）y-g_n（x），\ end {aligned} $$，其中\（f_m（x）\）和\（g_n（x）\）分别是次数为m的实多项式和n。论文的主要结果如下：我们给出了H（m，  n）。事实证明，在大多数情况下，这些界限是尖锐的。此外，还明确描述了超椭圆极限环的配置。