On the number of hyperelliptic limit cycles of Li\'{e}nard systems

In this paper, we study the maximum number, denoted by $H(m,n)$, of hyperelliptic limit cycles of the Li\'enard systems $$\dot x=y, \qquad \dot y=-f_m(x)y-g_n(x),$$ where, respectively, $f_m(x)$ and $g_n(x)$ are real polynomials of degree $m$ and $n$, $g_n(0)=0$. The main results of the paper are as follows: In term of $m$ and $n$ of the system, we obtain the upper bound and lower bound of $H(m,n)$ in all the possible cases. Furthermore, these upper bound can be reached in some cases.