On Melnikov functions of a homoclinic loop through a nilpotent saddle for planar near-Hamiltonian systems

Abstract

The first-order Melnikov function of a homoclinic loop through a nilpotent saddle for general planar near-Hamiltonian systems is considered. The asymptotic expansion of this Melnikov function and formulas for its first coefficients are given. The number of limit cycles which appear near the homoclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. An example is presented as an application of the main results.

On the number of limit cycles near a homoclinic loop with a nilpotent singular point

“…When β l > 2, it is difficult to find the value of coefficient of the corresponding term now. Similar problem also appeared in [28,30,31,53].…”

“…Zang, M. Han, D. Xiao [53] and M. Han, J. Yang and D. Xiao [29] studied the expansion ofM in the case when (1.2) has a homoclinic loop passing through a nilpotent saddle, see Fig. 1 (c).…”

“…Moreover, they obtained results about the existence of limit cycles by the expansion of Melnikov function as S 0 is a nilpotent cusp or nilpotent saddle, see [31] and [53].…”

Limit cycles in a Liénard system with a cusp and a nilpotent saddle of order 7

“…The coefficients c 2 ; c 3 ; c 5 ; c 6 and c 7 are called local coefficients of system (3) at the nilpotent saddle point (see [25]). …”

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

“…On the other hand, we consider the asymptotic expansion of I(h) as h → 1 − . The asymptotic expansion has been given by Example 11 in [28]. We recall it as follows.…”