For the Liénard systems with fm and gn polynomials of degree m and n, respectively, we present explicit systems having algebraic limit cycles in the cases m ≥ 2 and n ≥ 2m + 1 and m ≥ 3 and n = 2m. Also we prove that the Liénard system for m = 3 and n = 5 has no hyperelliptic limit cycles. This shows that the result of theorem 1(c) of Zoladek (1998 Trans. Am. Math. Soc. 350 1681-701) on the existence of algebraic limit cycles of the Liénard system is not correct. Moreover, we characterize all hyperelliptic limit cycles of the Liénard systems for m = 4 and n = 6 or n = 7. For m > 4 and n = 2m - 1 or n = 2m - 2 we prove that there are Liénard systems which have [m/2] - 1 algebraic limit cycles, where the  denotes the integer part function. © 2008 IOP Publishing Ltd and London Mathematical Society.
|Publication status||Published - 1 Sep 2008|