The hyperelliptic limit cycles of the Liénard systems

Abstract

For Liénard systemsẋ = y,ẏ = − 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 < 2m 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 < 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.

Liouvillian Integrability Versus Darboux Polynomials

“…In general Darboux integrability depends on the existence of Darboux polynomials and exponential factors. There are plenty of results studying the existence of Darboux polynomials, see for example [1,14,23,24,32,33,34,40] and the references mentioned there.…”

On the Number of Hyperelliptic Limit Cycles of Liénard Systems

“…In this section, we shall collect some related properties of Liénard systems and introduce a complete discrimination system for polynomials. For the proof of these results, we refer the reader to ( [3,4,9,10,11,12,13])for details.…”

“…An individual type (5, 7) of the Liénard system is discussed in [9], where Yu and Zhang clarified that there exist Liénard systems of (5, 7)-type which have hyperelliptic limit cycles.…”

On the hyperelliptic limit cycles of Liénard systems

“…Does a Liénard system (2) with m > 5 and m + 1 < n < 2m have an algebraic limit cycle?In this paper, by developing the main ideas in[3,8,9] we prove the following result, which gives a positive answer to the above open problem.…”

“…In this section, we shall collect some related known results. For the detailed proof of these results, we refer the reader to [3,8,9]. However, for the convenience of the reader, below we give an outline of some proofs.…”