Reply to “Comment on ‘Liénard systems, limit cycles, and Melnikov theory’ ”

Abstract

The problem of finding the number of limit cycles of Lienard systems, ẋ5y2«F(x ,m), ẏ52x , where F(x ,m) is an odd polynomial, was addressed by Giacomini and Neukirch @Phys. Rev. E 56, 3809 ~1998!#, and they proposed an original method, where a sequence of polynomials is introduced, whose roots give the number of limit cycles that also allow one to construct a sequence of algebraic approximations to the limit cycles. This author showed @Phys. Rev. E 57, 340 ~1998!# that in the limit of these sequences, the same information is given by a polynomial which Melnikov theory associates to each Lienard system. In their comment, Giacomini and Neukirch @Phys. Rev. E 59, 2483 ~1999!# remark that this is correct only for small values of « . I wish to stress here that this is right and that the reason for it lies in the perturbative nature of Melnikov theory, while the Giacomini and Neukirch method is nonperturbative. As a consequence the original conjecture is reformulated. @S1063-651X~98!12612-0#