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 ofdegree m and n respectively, Zoladek [Trans. Amer. Math. Soc. 350 (1998), 1681-1701]showed that if m≥3 and m + 1 < n < 2m there always exist Liénard systems whichhave a hyperelliptic limit cycle. Llibre and Zhang [Nonlinearity 21 (2008), 2011-2022]have proved that the Liénard systems with m = 3 and n = 5 have no hyperelliptic limitcycles and that there exist Liénard systems with m = 4 and 5 < n < 8 which do havehyperelliptic limit cycles. So, it is still an open problem to characterize the Liénardsystems having an algebraic limit cycle in cases m > 4 and m + 1 < n < 2m.In this paper we will prove that there exist Liénard systems with m = 5 and m +1 < n < 2m which have hyperelliptic limit cycles. Furthermore, for completenesswe characterize all Liénard systems with m = 5 and m + 1 < n < 2m which have ahyperelliptic limit cycle.