Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory

Consider the class of reversible quadratic systems (x) over dot = y, (y) over dot = -x + x(2) + y(2) - r(2), with r > 0. These quadratic polynomial differential systems have a center at the point ((1 - root 1 + 4r(2))/2, 0) and the circle x(2) + y(2) = r(2) is one of the periodic orbits surrounding this center. These systems can be written into the form (x) over dot = y + (4 + A)x(2) - Ay(2), (y) over dot = -x, with A is an element of (-2, 0). For all A is an element of R we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bifurcating from the periodic orbits surrounding the center (0, 0) of that system. Up to now this result was only known for A = -2 (see [Li, 2002; Liu, 2012]).