Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order \(m\) (Q1993971)

The authors study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order \(m\) in a near-Hamiltonian system. For \(m \geq 1\), they derive the formulas of the coefficients in the expansion, which can be used to study limit cycle bifurcations. In particular, for \(m = 2\), these coefficients are used to consider the limit cycle bifurcations of general near-Hamiltonian systems. Further, they give the existence conditions and the distributions for \(10, 11, 13, 15\) and \(16\) (respectively \(11, 13\) and \(16\)) limit cycles when the homoclinic loop is of cuspidal type (respectively smooth type). As an application, they consider a near-Hamiltonian system with a nilpotent saddle of order \(2\) and obtain the lower bounds of the maximal number of limit cycles.