Heteroclinic bifurcation near a loop tangent to an invariant line

In this paper, we propose a method for examining the heteroclinic bifurcation near a loop tangent to an invariant line in near-Hamiltonian systems. Our objective is to derive the asymptotic expansion of a generalized Melnikov function, which encompasses not only the first-order Melnikov function but also higher-order Melnikov functions in a wider range of reversible Hamiltonian systems. We apply our findings to a cubic reversible Hamiltonian system with polynomial perturbations of degree n. Our contributions include:

(i) Determining the precise number of limit cycles near the tangent loop by using the first-order Melnikov function for polynomial perturbations of arbitrary degree n.

(ii) Deriving all-order Melnikov functions with simplified expressions and integrable conditions for the system under the cubic perturbation ($n=3$). Our analysis reveals that the first, second, third, and fourth-order Melnikov functions lead to the bifurcation of $three$, $five$, six limit cycles, and one limit cycle near the loop, respectively.

(iii) Determining the exact upper bound on the maximum number of zeros of the first-order Melnikov function for the cubic perturbation by applying a modified Chebyshev criterion and an element-combination technique.