The hyperelliptic limit cycles of the Liénard systems

For Liénard systems x ˙ = y , y ˙ = − f m ( x ) y − g n ( x ) with f m and g n real polynomials of degree m and n respectively, in [H. Zoladek, Algebraic invariant curves for the Liénard equation, Trans. Amer. Math. Soc. 350 (1998) 1681–1701] the author showed that if m ⩾ 3 and m + 1 < n < 2 m there always exist Liénard systems which have a hyperelliptic limit cycle. Llibre and Zhang [J. Llibre, Xiang Zhang, On the algebraic limit cycles of Liénard systems, Nonlinearity 21 (2008) 2011–2022] proved that the Liénard systems with m = 3 and n = 5 have no hyperelliptic limit cycles and that there exist Liénard systems with m = 4 and 5 < n < 8 which do have hyperelliptic limit cycles. So, it is still an open problem to characterize the Liénard systems which have an algebraic limit cycle in cases m > 4 and m + 1 < n < 2 m . In this paper we will prove that there exist Liénard systems with m = 5 and m + 1 < n < 2 m which have hyperelliptic limit cycles.