Perturbations of non-Hamiltonian reversible quadratic systems with cubic orbits

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) M1(h) is not identically zero, then the perturbed system has at most 5 for n=3 and 3n-7 for n⩾4 limit cycles on the finite plane. If M1(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 6n-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.