The cyclicity of the period annulus of a reversible quadratic system

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system (x) over dot= y + ax(2), (y) over dot = -x with a not equal 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycles, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods cannot be applied, except in Figuerasa, Tucker and Villadelprat (2013, J. Diff. Equ., 254, 3647-3663) a computer-assisted method was used. In this paper, we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by Dumortier and Roussarie (2009, Discrete Contin. Lyn. Syst., 2, 723-781) for q <= 2. The method may be used in other problems.