Melnikov Functions in Quadratic Perturbations of Generalized Lotka–Volterra Systems

Abstract

We present a detailed analysis of Melnikov functions which arise in quadratic perturbations of generalized Lotka---Volterra vector fields with the first integral x?ys(1 ? x ? y). That analysis was sketched in ?ola?dek (J Differ Equ 109:223---273, 1994). In particular, we prove that the maximal number of limit cycles in the generic case equals 2 and in the Hamiltonian triangle case, this number is 3.