On the Melnikov functions and limit cycles near a double homoclinic loop with a nilpotent saddle of order [formula omitted

Abstract

For a centrally symmetric near-Hamiltonian system, we develop a method for computing all the coefficients in the expansions of three Melnikov functions near a double homoclinic loop. Moreover, we give a new estimation on the lower bound of H(2nˆ,5) for 11≤nˆ≤23, where H(2nˆ,5) is the maximal number of limit cycles for a kind of Liénard system, x˙=y,y˙=−g(x)+εf(x)y, with deg⁡g(x)=5 and deg⁡f(x)=2nˆ.