The cyclicity of the period annulus of a reversible quadratic system

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