Second-order bifurcation of limit cycles from a quadratic reversible center

This article concerns the bifurcation of limit cycles from a quadratic integrable and non-Hamiltonian system. By using the averaging theory, we show that under any small quadratic homogeneous perturbation, there is at most one limit cycle for the first order bifurcation and two for the secondorder bifurcation arising from the period annulus of the unperturbed system, respectively. Moreover, in each case the upper bound is sharp.