Perturbations of non-Hamiltonian reversible quadratic systems with cubic orbits

Abstract

This paper is concerned with degree n polynomial perturbations of a class of planar non-Hamiltonian reversible quadratic integrable system whose almost all orbits are cubics. We give an estimate of the number of limit cycles for such a system. If the first-order Melnikov function (Abelian integral) M 1 ( h ) is not identically zero, then the perturbed system has at most 5 for n = 3 and 3 n - 7 for n ⩾ 4 limit cycles on the finite plane. If M 1 ( h ) is identically zero but the second Melnikov function is not, then an upper bound for the number of limit cycles on the finite plane is 11 for n = 3 and 6 n - 13 for n ⩾ 4 , respectively. In the case when the perturbation is quadratic ( n = 2 ), there exists a neighborhood U of the initial non-Hamiltonian polynomial system in the space of all quadratic vector fields such that any system in U has at most two limit cycles on the finite plane. The bound for n = 2 is exact.