On the Formal Integrability Problem for Planar Differential Systems

and Applied Analysis 3 4. The ε-Method for Resonant Centers We recall in this section the ε-method developed in [23] which we apply here to resonant centers. We consider system (2) which we write into the form ?̇? = P (x, y) = px + F 1 (x, y) , ?̇? = Q (x, y) = −q y + F 2 (x, y) , (5) where F 1 (x, y) and F 2 (x, y) are analytic functions without constant and linear terms, defined in a neighborhood of the origin. To implement the algorithm we introduce a rescaling of the variables and a time rescaling given by (x, y, t) → (ε p x, ε q y, ε r t) where ε > 0 and p, q, and r ∈ Z and system (5) takes the form ?̇? = ε r−p (p ε p x + F 1 (ε p x, ε q y)) , ?̇? = ε r−q (−q ε q y + F 2 (ε p x, ε q y)) . (6) We choose p, q, r in such a way that system (6) will be analytic in ε. Hence, by the classical theorem of the analytic dependence with respect to the parameters, we have that system (6) admits a first integral which can be developed in power of series of ε because it is analytic with respect to this parameter. Therefore, we can propose the following development for the first integral: