On the upper bound of the number of limit cycles obtained by the second order averaging method

For ε small we consider the number of limit cycles of the system ẋ = -y(1 + x) + εF(x, y), ẏ = x(1 + x) + εG(x, y), where F and G are polynomials of degree n starting with terms of degree 1. We prove that at most 2n - 1 limit cycles can bifurcate from the periodic orbits of the unperturbed system (ε = 0) using the averaging theory of second order under the condition that the second order averaging function is not zero. Copyright ©2007 Watam Press.